Chapter 6 Performance Tasks - BergSSMathbergssmath.weebly.com/uploads/4/2/4/6/4246450/6-4...[2] Correct answer with no work shown OR student only able to complete part of Task. [1]

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Name Class Date

Chapter 6 Performance Tasks

Give complete answers.

Task 1

a. Write a product of two square roots so that the answer, when simplified is 12x3y2. Show how your product simplifies to give the correct answer.

b. Write a quotient of two cube roots so that the answer, when simplified, is 3a2

4b3. Show how your quotient simplifies to give the correct answer.

c. Write a product of the form (a + 1b)(a - 1b) so that the answer, when simplified, is 59. Show how your product simplifies to give the correct answer.

Task 2

a. Find a radical equation of the form 1ax + b = x + c so that one solution is extraneous. Show the steps in solving the equation.

b. Is there a value for h that makes it possible for the equation 1x + h + 5 = 0 to have any real number solutions? Explain.

c. Explain the relationship between the solutions to the equation 1x - 3 - 2 = 0 and the graph of the function y = 1x - 3 - 2.

[4] Check students’ work. All parts of Task completed correctly with work shown OR used correct process with minor computational errors.

[3] Student found products and quotient correctly but followed through the process incorrectly using incorrect simplification.

[2] Correct answer with no work shown OR student only able to complete part of Task.

[1] Student understood that squaring, cubing or multiplying was involved but was unable to correctly find the products or quotient, or to simplify them.

[0] No attempt was made to solve this problem OR answer is incorrect with no work shown

[4] Check students’ work. All parts of Task completed correctly with work shown OR used correct process with minor computational errors.

[3] Student found parts (a) and (b) correctly but could not explain (c) OR could not show steps in (a) OR could not explain (b).


[1] Student know to square each side of the equation, but could not complete the solution.

[0] No attempt was made to solve this problem OR answer is incorrect with no work shown.


Name Class Date

Chapter 6 Performance Tasks (continued)

Let f (x) = x2 + x − 12 and g(x) = x − 2. Answer each of the following questions.

Task 3

a. Find g(x)f (x) and its domain. Explain how you determined the domain.

b. Find (g ∘ f )(x) and ( f ∘ g)(x). Are they equal?

c. For what types of functions will (g ∘ f )(x) and ( f ∘ g)(x) both equal x? Explain.

Give complete answers.

Task 4

a. Find the inverse of f (x) = 1x - 2 + 5. Show all steps in the process. What

is the domain of f -1?

b. Choose a value for a and use the inverse to find (f ∘ f -1)(a) and

(f -1 ∘ f)(a) for the value you chose. What can you conclude about

(f ∘ f -1)(a) and (f -1 ∘ f)(a)?

c. Graph f and f -1 on the same axes. What relationships do you see between the two graphs?

[4] All parts of Task completed correctly with work shown OR used correct process with minor computational errors.

[3] Student found all parts correctly but could not explain (a) OR could not explain (c) OR could not correctly find the domain for (a).


[1] Student found part (a) but could not compose functions for parts (b) and (c).[0] No attempt was made to solve this problem OR answer is incorrect with no

work shown.

[4] All parts of Task completed correctly with work shown OR used correct process with minor computational errors.

[3] Student found parts (a) and (b) correctly but could not describe the relationships between the graphs OR could not draw conclusions in part (b).


[1] Student found the inverse of f(x) but could not draw conclusions in part (b) and could not graph the functions.

[0] No attempt was made to solve this problem OR answer is incorrect with no work shown.

x − 2x2 + x − 12

; all real numbers

except 3 and −4x2 + x − 14; x2 − 3x − 10; no

when f and g are inverses

The graphs are a reflection of each other across the line y = x.

f−1(x) = (x − 5)2 + 2; domain: x # 5

(f 5 f−1)(a) = (f−1 5 f)(a) = aO

2

4

6

62 4

x

y f

f�1


Name Class Date

Additional Vocabulary Support Roots and Radical Expressions

Complete the vocabulary chart by filling in the missing information.

Word or Word Phrase

Definition Example

nth root Given the equation an = b, a is the nth root of b.

1.

radicand 2. The radicand in the expression 13 64 is 64.

index The number that gives the degree of the root.

3.

cube root The third root of a number. 4.

principal root 5. The principal square root of 4 is 2.

Choose the word or phrase from the list that best completes each sentence.

cube root nth root radicand index principal root

6. The is the number under the radical sign in a radical expression.

7. The of 27 is 3.

8. Given the equation an = b, a is the of b.

9. In a radical expression, the indicates the degree of the root.

10. When a number has both a positive and a negative root, the positive root is

considered the .

34 = 81; 3 is the 4th root of 81.

The cube root of 8 is 2.

The index in the

expression 5132 is 5.

The number under the radical sign.

The positive root when a number has both a positive and a negative root.

radicand

cube root

nth root

index

principal root


Name Class Date

Think About a PlanRoots and Radical Expressions

Boat Building Boat builders share an old rule of thumb for sailboats. The maximum speed K in knots is 1.35 times the square root of the length L in feet of the boat’s waterline. a. A customer is planning to order a sailboat with a maximum speed

of 12 knots. How long should the waterline be? b. How much longer would the waterline have to be to achieve a maximum

speed of 15 knots?

1. Write an equation to relate the maximum speed K in knots to the length L in feet of a boat’s waterline.

2. How can you find the length of a sailboat’s waterline if you know its maximum speed?

.

3. A customer is planning to order a sailboat with a maximum speed of 12 knots. How long should the waterline be?

4. How can you find how much longer the waterline would have to be to achieve a maximum speed of 15 knots, compared to a maximum speed of 12 knots?

.

5. If a customer wants a sailboat with a maximum speed of 15 knots, how long should the waterline be?

6. How much longer would the waterline have to be to achieve a maximum speed of 15 knots?

K = 1.351L

Substitute the maximum speed for K and solve the resulting equation for L

Subtract the waterline length needed for a 12-knot maximum speed from the

waterline length needed for a 15-knot maximum speed

about 123 ft

about 44 ft

about 79 ft


Name Class Date

Practice Form G

Roots and Radical Expressions

Find all the real square roots of each number.

1. 400 2. -196 3. 10,000 4. 0.0625

Find all the real cube roots of each number.

5. 216 6. -343 7. -0.064 8. 100027

Find all the real fourth roots of each number.

9. -81 10. 256 11. 0.0001 12. 625

Find each real root.

13. 1144 14. - 125 15. 1-0.01 16. 13 0.001

17. 14 0.0081 18. 13 27 19. 13 -27 20. 10.09

Simplify each radical expression. Use absolute value symbols when needed.

21. 281x4 22. 2121y10 23. 23 8g6

24. 23 125x9 25. 25 243x5y15 26. 23 (x - 9)3

27. 225(x + 2)4 28. 53 64x9

343 29. 13 -0.008

30. 54 x4

81 31. 236x2y6 32. 24 (m - n)4

33. A cube has volume V = s3, where s is the length of a side. Find the side length for a cube with volume 8000 cm3.

34. The temperature T in degrees Celsius (°C) of a liquid t minutes after heating is given by the formula T = 81t . When is the temperature 48°C?

12

0.3

9x2

3xy3

6

no real fourth roots

3

11 ∣ y5 ∣

x − 9

−0.2

6 ∣ x ∣y3∣ x ∣3

∣m − n ∣

36 min

20 cm

no real square roots

not a real number

−3

2g2

5(x + 2)2

0.1

0.3

5x3

4x3

7

103

−20, 20

−5

−7

−4.4

−0.4

−100, 100

−0.1, 0.1

−0.25, 0.25

−5, 5


Name Class Date

Find the two real solutions of each equation.

35. x2 = 4 36. x4 = 81

37. x2 = 0.16 38. x2 = 1649

39. x4 = 16625 40. x2 = 121

625

41. x2 = 0.000009 42. x4 = 0.0001

43. The number of new customers n that visit a dry cleaning shop in one year is directly related to the amount a (in dollars) spent on advertising. This relationship is represented by n3 = 13,824a. To attract 480 new customers, how much should the owners spend on advertising during the year?

44. Geometry The volume V of a sphere with radius r is given by the formula V = 4

3 pr3. a. What is the radius of a sphere with volume 36p cubic inches? b. If the volume increases by a factor of 8, what is the new radius?

45. A clothing manufacturer finds the number of defective blouses d is a function of the total number of blouses n produced at her factory. This function is d = 0.000005n2.

a. What is the total number of blouses produced if 45 are defective? b. If the number of defective blouses increases by a factor of 9, how does the

total number of blouses change?

46. The velocity of a falling object can be found using the formula v2 = 64h, where v is the velocity (in feet per second) and h is the distance the object has already fallen.

a. What is the velocity of the object after a 10-foot fall? b. How much does the velocity increase if the object falls 20 feet

rather than 10 feet?

Practice (continued) Form G


−2, 2

$8000

3 in.

6 in.

3000

about 25.30 ft/sec

It has tripled.

about 10.48 ft/sec

−0.003, 0.003

−25, 25

−3, 3

−0.1, 0.1

−0.4, 0.4

−1125, 11

25

−47, 47


Name Class Date

Find all the real square roots of each number.

1. 625 2. -1.44 3. 1681

Find all the real cube roots of each number.

4. -216 5. 164 6. 0.027

Find all the real fourth roots of each number.

7. 0.2401 8. 1 9. -1296

Find each real root. To start, fi nd a number whose square, cube, or fourth is equal to the radicand.

10. 1400 11. - 14 256 12. 13 -729

= 2(20)2

Simplify each radical expression. Use absolute value symbols when needed. To start, write the factors of the radicand as perfect squares, cubes, or fourths.

13. 225x6 14. 23 343x9y12 15. 24 16x16y20

= 2(5)2(x3)2

Practice Form K


t25

−6

t0.7

20

5 ∣ x 3 ∣

no real roots

14

t1

−4

7x 3y 4

t 49

0.3

no real fourth roots

−9

2x 4 ∣ y 5 ∣


Name Class Date

16. The formula for the volume of a sphere is V = 43 pr3. Solving for r, the radius of

a sphere is r = 53 3V4p. If the volume of a sphere is 20 ft3, what is the radius of the

sphere to the nearest hundredth?

Find the two real solutions of each equation.

17. x4 = 81 18. x2 = 144 19. x4 = 2401625

20. Writing Explain how you know whether or not to include the absolute value symbol on your root.

21. Arrange the numbers 13 -64, - 13 -64, 164, and 16 64, in order from least to greatest.

22. Open-Ended Write a radical that has no real values.

23. Reasoning There are no real nth roots of a number b. What can you conclude about the index n and the number b?

Practice (continued) Form K


t3 t12 t75

1.68 ft

If the index is odd, then you do not use the absolute value symbol on your root. If the index is even, then you need the absolute value symbol on those variable terms with an odd power.

Answers may vary. Sample: any even index radical with a negative radicand

The index n is even and the number b is negative.

31−64, 6164, − 31−64, 164


Name Class Date

Multiple Choice

For Exercises 1−6, choose the correct letter.

1. What is the real square root of 0.0064?

0.4 0.04

0.08 no real square root

2. What is the real cube root of -64?

4 -8

-4 no real cube root

3. What is the real fourth root of - 1681?

23 -4

9

-23 no real fourth root

4. What is the value of 13 -0.027?

-0.3 0.3 -0.03 0.03

5. What is the simplified form of the expression 24x2y4?

2xy2 2 0 x 0 y2 4xy2 2 0 xy 0

6. What are the real solutions of the equation x4 = 81?

-9, 9 3 -3, 3 -3

Short Response

7. The volume V of a cube with side length s is V = s3. A cubical storage bin has volume 5832 cubic inches. What is the length of the side of the cube? Show your work.

Standardized Test Prep Roots and Radical Expressions

B

G

D

F

B

H

[2] V = s3, 5832 = s3, s = 315832 = 18; 18 in.[1] incorrect side length OR no work shown[0] incorrect answer and no work shown OR no answer given


Name Class Date

Rounding Roots and RadicalsComputers treat radicals such as 12 as if they were rounded to a preassigned number of decimal places. Most computers round numbers according to an algorithm that uses the largest integer less than or equal to a given number. This function is called the greatest integer function and is written as y = [x].

As you can see, the graph of the greatest integer function is not continuous. The open circles indicate that the endpoint is not included as part of the graph.

The command INT in most popular spreadsheet programs serves the same purpose as the greatest integer function. For instance, INT(3.84) = 3; INT(-1.99) = -2; INT(7) = 7.

To round a number x to r decimal places, a computer performs the following procedure:

Step 1 Multiply x by 10r.

Step 2 Add 0.5 to the result.

Step 3 Find INT of the result.

Step 4 Multiply the result by 10-r.

Fill in the table below to see how this procedure works.

x r Step 1 Step 2 Step 3 Step 4

11.4825 3 11482.5 11483

132.718 2

34.999 1

A computer that rounds numbers after each operation may introduce rounding errors into calculations. To see the effects of rounding errors, perform each of the following computations for x = 2 and different r values. First find the given root and write the answer to r + 1 digits after the decimal. Carry out the four steps to get the answer and then raise the result to the given power. Write the answer again to r + 1 digits after the decimal and carry out the four steps to get the final answer.

x r (1x)2 (13 x )3

2 6

2 3

2 1

EnrichmentRoots and Radical Expressions

O

2

�2 2

x

y

2.000001

1.999

2.0

2.000000

2.000

2.2

11483

13271.8

349.99

13272

350

13272.3

350.49

132.72

35.0

11.483


Name Class Date

For any real numbers a and b and any positive integer n, if a raised to the nth power equals b, then a is an nth root of b. Use the radical sign to write a root. The following expressions are equivalent:

an = b · 1n b = a

Problem

What are the real-number roots of each radical expression?

a. 13 343 Because (7)3 = 343, 7 is a third (cube) root of 343. Therefore, 31343 = 7. (Notice that (-7)3 = -343, so -7 is not a cube root of 343.)

b. 54 1625 Because 11

524 = 1625 and 1 -1

524 = 1625, both 15 and - 15 are real-number fourth roots of 1

625.

c. 13 -0.064 Because (-0.4)3 = -0.064,-0.4 is a cube root of -0.064 and is, in fact, the only one. So, 31-0.064 = -0.4.

d. 1-25 Because (5)2 = (-5)2 = 25, neither 5 nor -5 are second (square) roots of -25. There are no real-number square roots of -25.

Exercises

Find the real-number roots of each radical expression.

1. 1169 2. 13 729 3. 14 0.0016

4. 53 - 18 5. 5 4121 6. 53 125

216

7. 5 - 425 8. 14 0.1296 9. 13 -0.343

10. 14 -0.0001 11. 55 1243 12. 53 8

125

Reteaching Roots and Radical Expressions

power index radicand

radical sign

−13, 13

−12

no real sq root

no real 4th root

9

− 211, 2

11

−0.6, 0.6

13

−0.2, 0.2

56

−0.7

25


Name Class Date

You cannot assume that 2nan = a. For example, 2(-6)2 = 136 = 6, not -6. This

leads to the following property for any real number a:

If n is odd 2nan = a

If n is even 2nan = 0 a 0

Problem

What is the simplified form of each radical expression?

a. 23 1000x3y9

23 1000x3y9 = 23 103x3(y3)3 Write each factor as a cube.

= 23 (10xy3)3 Write as the cube of a product.

= 10xy3 Simplify.

b. 54 256g8

h4k16

Write each factor as a power of 4.

= 54

a4g2

hk4b4

Write as the fourth power of a quotient.

= 4g2

0 h 0 k4 Simplify.

The absolute value symbols are needed to ensure the root is positive when h is negative. Note that 4g2 and k4 are never negative.

Exercises

Simplify each radical expression. Use absolute value symbols when needed.

13. 236x2 14. 23 216y3 15. 5 1100x2

16. 2x202y8 17. 53 (x + 3)3

(x - 4)6 18. 25 x10y15z5

19. 53 27z3

(z + 12)6 20. 24 2401x12 21. 53 1331x3

22. 54 (y - 4)8

(z + 9)4 23. 53 a6b6

c3 24. 23 -x3y6

Reteaching (continued) Roots and Radical Expressions

54 44(g2)4

h4(k4)4=54 256g 8

h4k16

6 ∣ x ∣

x10

y4

3z(z + 12)2

(y − 4)2

∣ z + 9 ∣

6y

x + 3(x − 4)2

7 ∣ x3 ∣

a2b2c

110 ∣ x ∣

x2y3z

11x

−xy2


Name Class Date

Additional Vocabulary Support Multiplying and Dividing Radical Expressions

Combining Radicals: Products

If 1n a and 1n b are real numbers, then 1n a # 1n b = 1n ab.

Sample 13 8 # 13 27 = 13 8 # 27 = 13 216 = 6

Solve.

1. 13 16 # 13 4 =

2. Which of the following products can be simplified? Circle the correct answer.

13 12 # 16 14 16 # 14 24 14 35 # 13 10

3. Write the radical expression 13 32x4 in simplest form.

4. Which of the following products cannot be simplified? Circle the correct answer.

14 15 # 14 4 14 # 112 14 10 # 13 5

5. 24x2y3 # 227x2y2 =

Combining Radicals: Quotients

If 1n a and 1n b are real numbers and b ≠ 0, then 1n a1n b= 1n a

b.

Sample 1812= 1812

= 14 = 2

Solve.

6. Which of the following quotients can be simplified? Circle the correct answer.

13 1213 4 13 623

14 2013 15

7. Write the radical expression 264x424x2 in simplest form.

8. Rewriting an expression so that there are no radicals in any denominator and no

denominators in any radical is called .

3116 # 4 = 3164 = 4

314x

6x2y223y

|4x|

rationalizing the denominator

2x


Name Class Date

Think About a Plan Multiplying and Dividing Radical Expressions

Satellites The circular velocity v, in miles per hour of a satellite orbiting Earth is

given by the formula v = 51.24 * 1012r , where r is the distance in miles from

the satellite to the center of the Earth. How much greater is the velocity of a satellite orbiting at an altitude of 100 mi than the velocity of a satellite orbiting at an altitude of 200 mi? (The radius of the Earth is 3950 mi.)

Know

1. The first satellite orbits at an altitude of e e.

2. The second satellite orbits at an altitude of e e.

3. The distance from the surface of the Earth to its center is e e.

Need

4. To solve the problem I need to find:

.

Plan

5. Rewrite the formula for the circular velocity of a satellite using a for the altitude of the satellite.

6. Use your formula to find the velocity of a satellite orbiting at an altitude of 100 mi.

7. Use your formula to find the velocity of a satellite orbiting at an altitude of 200 mi.

8. How much greater is the velocity of a satellite orbiting at an altitude of 100 mi than one orbiting at an altitude of 200 mi?

100 mi

200 mi

the difference in the velocities of a satellite orbiting at an altitude of 100 mi

and one orbiting at an altitude of 200 mi

v = 51.24 × 1012

a + 3950

about 17,498 mi/h

about 17,286 mi/h

about 212 mi/h

3950 mi


Name Class Date

Multiply, if possible. Then simplify.

1. 14 # 125 2. 181 # 136 3. 13 # 13 27

4. 23 45 # 23 75 5. 118 # 150 6. 13 -16 # 13 4

Simplify. Assume that all variables are positive.

7. 236x3 8. 23 125y2z4 9. 218k6

10. 23 -16a12 11. 2x2y10z 12. 24 256s7t12

13. 23 216x4y3 14. 275r3 15. 24 625u5v8

Multiply and simplify. Assume that all variables are positive.

16. 14 # 16 17. 29x2 # 29y5 18. 23 50x2z5 # 23 15y3z

19. 412x # 318x 20. 1xy # 14xy 21. 912 # 31y

22. 212x2y # 23xy4 23. 23 -9x2y4 # 23 12xy 24. 723y2 # 226x3y

Divide and simplify. Assume that all variables are positive.

25. 17513 26.

163xy317y 27.

154x5y312x2y

28. 16x13x 29.

314x231x

30. 24 243k3

3k7

31. 2(2x)21(5y)4 32.

3218y2

3112y 33. 5162a

6a3

Practice Form G

Multiplying and Dividing Radical Expressions

15 30 −4

10

226

6xy22xy −3xy 324y2 42xy22xy

54

9xy21y

9

5yz2 326x2

6x1x 5z 32y2z 3k312

−2a4 312 xy51Z 4st3 41s3

6xy 31x 5r23r 5uv2 41u

48x 2xy 2712y

5 3y1x 3xy13x

323a

22324x 3

k

2x25y2

2312y2


Name Class Date

Rationalize the denominator of each expression. Assume that all variables are positive.

34. 1y15

35. 118x2y12y3 36.

137xy2134x2

37. 59x2 38.

1xy13x 39. 53 x2

3y

40. 14 2x143x2

41. 5 x8y 42. 53 3a

4b2c

43. What is the area of a rectangle with length 1175 in. and width 163 in.?

44. The area of a rectangle is 30 m2. If the length is 175 m, what is the width?

45. The volume of a right circular cone is V = 13pr2h, where r is the radius of the base and h

is the height of the cone. Solve the formula for r. Rationalize the denominator.

46. The volume of a sphere of radius r isV = 43pr3.

a. Use the formula to find r in terms of V. Rationalize the denominator. b. Use your answer to part (a) to find the radius of a sphere with volume

100 cubic inches. Round to the nearest hundredth.

Simplify each expression. Rationalize all denominators. Assume that all variables are positive.

47. 114 # 121 48. 13 150 # 13 20 49. 13(112 - 16)

50. 612x513

51. 8132x2

52. 513

xy41325xy2



25y5

3xy

23 14x2y2

2x

322x224 54x3

3x

23y322xy

4y

23 9x2y2

3y23 6abc2

2bc

105 in.2

213 m

r = 23PhVPh

2.88 in.

726 6 − 322

226x5

4 324xx

3 25y2

10 323

r = 323V4P

; r =326P2V

2P


Name Class Date

Practice Form K


Multiply, if possible. Then simplify. To start, identify the index of each radical.

1. 13 4 # 13 6 2. 15 # 18 3. 13 6 # 14 9

index of both radicals is 313 4 # 6

Simplify. Assume all variables are positive. To start, change the radicand to factors with the necessary exponent.

4. 23 27x6 5. 248x3y4 6. 25 128x2y25

= 23 33 # (x2)3

Multiply and simplify. Assume all variables are positive.

7. 112 # 13 8. 24 7x6 # 24 32x2 9. 223 6x4y # 323 9x5y2

Simplify each expression. Assume all variables are positive.

10. 13 4 # 13 80 11. 522xy6 # 222x3y 12. 15115 + 1152

13. Error Analysis Your classmate simplified 25x3 # 23 5xy2 to 5x2y. What mistake did she make? What is the correct answer?

14. A square rug has sides measuring 13 16 ft by 13 16 ft. What is the area of the rug?

2 3 13

3x 2

6

4 315

They are different, so you cannot multiply the radicands.

4 314 ft 2

2110

4xy 213x

2x 2 4114

20x 2y 31y

The indexes are different, so you cannot multiply.

2y 5 524x 2

18x 3y 312

5 + 513

She thought the indexes were the same.


Name Class Date

Divide and simplify. Assume all variables are positive. To start, write the quotient of roots as a root of a quotient.

15. 236x619x4 16. 24 405x8y214

5x3y2 17.

23 75x7y21325x4

= 536x6

9x4

Rationalize the denominator of each quotient. Assume all variables are positive. To start, multiply the numerator and denominator by the appropriate radical expression to eliminate the radical.

18. 12613 19. 13 x13 2

20. 27x4y15xy

= 12613# 1313

21. Einstein’s famous formula E = mc2 relates energy E, mass m, and the speed of light c. Solve the formula for c. Rationalize the denominator.

22. The formula h = 16t2 is used to measure the time t it takes for an object to free fall from height h. If an object falls from a height of h = 18a5 ft, how long did it take for the object to fall in terms of a?



2x

1783

c = 5Em ; c = 1Em

m

3a 212a4 seconds

3x 41x

314x2

x 323y 2

x 135x5


Name Class Date

Standardized Test Prep Multiplying and Dividing Radical Expressions

Multiple Choice

For Exercises 1−5, choose the correct letter. Assume that all variables are positive.

1. What is the simplest form of 13 -49x # 23 7x2?

7x17x -7x 7x -723 x2

2. What is the simplest form of 280x7y6?

2x3y3120x 4x6y625x3 425x7y6 4x3y315x

3. What is the simplest form of 23 25xy2 # 23 15x2?

5x23 3y2 5x13 3y 15xy13 y 5xy115x

4. What is the simplest form of 275x5112xy2?

523x4

213y2 5x2

2y 5x1x2y

5x2y2

5. What is the simplest form of 223 x2y13

4xy2 ?

23 x2y

2y x13 2y

y 23 2xy2

y 13 2yxy

Short Response

6. The volume V of a wooden beam is V = ls2, where l is the length of the beam and s is the length of one side of its square cross section. If the volume of the beam is 1200 in.3 and its length is 96 in., what is the side length? Show your work.

B

I

A

G

C

[2] V = ls2; s = 5Vl = 51200

96 = 112.5 ? 3.5 in.

[1] appropriate methods but with computational errors OR correct answer without work shown

[0] incorrect answer and no work shown OR no answer given


Name Class Date

EnrichmentMultiplying and Dividing Radical Expressions

To simplify the radical 1n a, you look for a perfect nth power among the factors of the radicand a. When this factor is not obvious, it is helpful to factor the number into primes. Prime numbers are important in many aspects of mathematics. Several mathematicians throughout history have unsuccessfully tried to find a pattern that would generate the nth prime number. Other mathematicians have offered conjectures about primes that remain unresolved.

1. Goldbach’s Conjecture states that every even number n 7 2 can be written as the sum of two primes. For example, 4 = 2 + 2 and 10 = 3 + 7. Choose three even numbers larger than 50 and write them as a sum of two primes.

2. The Odd Goldbach’s Conjecture states that every odd number n 7 5 can be written as the sum of three primes. For example, 7 = 2 + 2 + 3. Choose three odd numbers larger than 50 and write them as the sum of three primes.

3. Another interesting pattern emerges when you examine a subset of the prime numbers. Make a list of the primes less than 50.

4. Make this list smaller by eliminating 2 and all primes that are 1 less than a multiple of 4.

5. The remaining primes in the list above are related in an interesting way. You can write each prime as the sum of two squares. Express each of these primes as a sum of two squares.

6. A Cullen number, named after an Irish mathematician James Cullen, is a natural number of the form n * 2n + 1. Determine the first four Cullen numbers. That is, let n = 1, 2, 3, 4.

7. What is the smallest Cullen number that is a prime number? (The next Cullen number that is a prime occurs when n = 141!)

8. A palindrome is a number that reads the same forward and backward. For example, 121 is a palindromic number. List the seven palindromic primes that are less than 140.

Answers may vary. Sample: 52 = 47 + 5.

Answers may vary. Sample: 51 = 37 + 11 + 3.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

5, 13, 17, 29, 37, 41

5 = 1 + 4, 13 = 4 + 9, 17 = 1 + 16, 29 = 4 + 25, 37 = 1 + 36, 41 = 16 + 25

3, 9, 25, 65

3 when n = 1

2, 3, 5, 7, 11, 101, 131


Name Class Date

ReteachingMultiplying and Dividing Radical Expressions

You can simplify a radical if the radicand has a factor that is a perfect nth power and n is the index of the radical. For example:1n xynz = y1n xz

Problem

What is the simplest form of each product?

a. 13 12 # 13 10

13 12 # 13 10 = 13 12 # 10 Use

= 23 22 # 3 # 2 # 5 Write as a product of factors.

= 23 23 # 3 # 5 Find perfect third powers.

= 23 23 # 23 3 # 5 Use n1ab = n 1a # n1b .

= 213 15 Use n2an = a to simplify.

b. 27xy3 # 221xy2

27xy3 # 221xy2 = 27xy3 # 21xy2 Use n1a # n1b = n 1ab .

= 27xy2y # 3 # 7xy2 Write as a product of factors.

= 272x2(y2)2 # 3y Find perfect second powers.

= 7xy223y Use n1an = a to simplify.

Exercises

Simplify each product.

1. 115x # 135x 2. 2350y2 # 23

20y 3. 23 36x2y5 # 23 -6x2y

4. 527x3y # 228y2 5. - 23 9x5y2 # 232x2y5 6. 13 (112 - 121 )

n1a # n1b = n 1ab .

5x221 10y −6xy2 31x

70xy2xy −x2y2 3118xy 6 − 327


Name Class Date

Rationalizing the denominator means that you are rewriting the expression so that no radicals appear in the denominator and there are no fractions inside the radical.

Problem

What is the simplest form of 19y12x

?

Rationalize the denominator and simplify. Assume that all variables are positive.

19y12x

= 59y2x

Rewrite as a square root of a fraction.

= 59y # 2x2x # 2x

Make the denominator a perfect square.

= 518xy4x2

Simplify.

= 118xy122 # x2 Write the denominator as a product of perfect squares.

= 218xy2x Simplify the denominator.

= 232 # 2 # x # y2x Simplify the numerator.

= 322xy2x Use n1an = a to simplify.

Exercises

Rationalize the denominator of each expression. Assume that all variables are positive.

7. 151x 8. 23 6ab213

2a4b 9.

24 9y14 x 10.

210xy3112y2

11. 423 k9

1613k5

12. 53x5

5y 13. 24 1014

z2 14. 73 19a2b

abc4

Reteaching (continued)


25xx

323ba

429x3yx

230xy6

k 32k4

x2215xy5y

4210z2

z

3219ac2

c2


Name Class Date

The column on the left shows the steps used to rationalize a denominator. Use the column on the left to answer each question in the column on the right.

Problem Rationalizing the Denominator

Write the expression 41317 + 13 with a

rationalized denominator.

1. What does it mean to rationalize a denominator?

Multiply the numerator and the denominator by the conjugate of the denominator.

41317 + 13# 17 - 1317 - 13

2. What are conjugates?

The radicals in the denominator cancel out.

413117 - 1327 - 3

3. Write and solve an equation to show why the radicals in the denominator cancel out.

Distribute 13 in the numerator.

4(13 # 17 - 13 # 13)7 - 3

4. What property allows you to distribute the 13?

Simplify.

4(121 - 3)4

5. Why do the fours in the numerator and the denominator cancel out?

Simplify.121 - 3

6. What number multiplied by 121 would produce a product of 21?

Additional Vocabulary Support Binomial Radical Expressions

Sample answer: It means to write

an expression so that there are no

radicals in any denominators and no

denominators in any radicals.

Conjugates are expressions that differ

only in the signs of the first or second

terms.

(17 + 13)(17 − 13) =

(17 # 17) − (17 # 13) +

(17 # 13) − (13 # 13) = 7 − 3

The Distributive Property

Sample answer: Because 4 divided by

4 equals 1.

121


Name Class Date

Think About a PlanBinomial Radical Expressions

Geometry Show that the right triangle with legs of length 12 - 1 and 12 + 1 is similar to the right triangle with legs of length 6 - 132 and 2.

Understanding the Problem

1. What is the length of the shortest leg of the first triangle? Explain.

2. What is the length of the shortest leg of the second triangle? Explain.

3. Which legs in the two triangles are corresponding legs?

Planning the Solution

4. Write a proportion that can be used to show that the two triangles are similar.

Getting an Answer

5. Simplify your proportion to show that the two triangles are similar.

12 − 1 ; because 12 = 12, 12 − 1 must be less than 12 + 1

6 − 132; because 132 is between 5 and 6, 6 − 132 must be between 0 and 1,

which is less than 2.

The smaller leg in the first triangle corresponds to the smaller leg in the second

triangle. The larger leg in the first triangle corresponds to the larger leg in the

second triangle.

12 − 112 + 1≟ 6 − 132

2

12 − 112 + 1≟ 6 − 132

2

2(12 − 1)≟ (12 + 1)(6 − 132)

212 − 2≟ 612 − 164 + 6 − 132

212 − 2≟ 612 − 8 + 6 − 412

212 − 2 = 212 − 2


Name Class Date

Practice Form G

Binomial Radical Expressions

Add or subtract if possible.

1. 913 + 213 2. 512 + 213 3. 317 - 713 x

4. 1413 xy - 313 xy 5. 813 x + 213 y 6. 513 xy + 13 xy

7. 13x - 213x 8. 612 - 513 2 9. 71x + x17

Simplify.

10. 3132 + 2150 11. 1200 - 172 12. 13 81 - 313 3

13. 214 48 + 314 243 14. 3175 + 2112 15. 13 250 - 13 54

16. 128 - 163 17. 314 32 - 214 162 18. 1125 - 2120

Multiply.

19. 11 - 15212 - 152 20. 11 + 4110212 - 1102 21. 11 - 317214 - 3172

22. (4 - 213)2 23. (12 + 17)2 24. 1213 - 31222

25. 14 - 13212 + 132 26. 13 + 111214 - 1112 27. 1312 - 21322

Multiply each pair of conjugates.

28. (312 - 9)(312 + 9) 29. (1 - 17)(1 + 17)

30. (513 + 12)(513 - 12) 31. (312 - 213)(312 + 213)

32. (111 + 5)(111 - 5) 33. (217 + 313)(217 - 313)

8 31x + 2 31y

612 − 5 312

412

1913

0

−38 + 7210

9 + 2114

1 + 111

−13x

2212

13 413

−17

7 − 315

1113

28 − 1613

11 31xy

5 + 213

−63

73

−14

−6

6

1

6 31xy

71x + x17

0

2 312

15

67 − 1517

317 − 7 31x512 + 213

30 − 1216

30 − 1216


Name Class Date



Rationalize each denominator. Simplify the answer.

34. 3 - 11015 - 12 35. 2 + 11417 + 12

36. 2 + 13 x13 x

Simplify. Assume that all the variables are positive.

37. 128 + 4163 - 217 38. 6140 - 2190 - 31160

39. 3112 + 7175 - 154 40. 413 81 + 213 72 - 313 24

41. 31225x + 51144x 42. 6245y2 + 4220y2

43. 131y - 152121y + 5152 44. 11x - 13211x + 132 45. A park in the shape of a triangle has a sidewalk dividing it into two parts.

a. If a man walks around the perimeter of the park, how far will he walk? b. What is the area of the park?

46. The area of a rectangle is 10 in.2. The length is 12 + 122 in. What is the width?

47. One solution to the equation x2 + 2x - 2 = 0 is -1 + 13. To show this, let x = -1 + 13 and answer each of the following questions.

a. What is x2? b. What is 2x? c. Using your answers to parts (a) and (b), what is the sum x2 + 2x - 2?

600 ft

300 ft

30!0 ft

300 6 ft

side

wal

k !

300 3 ft!

300

3

ft!

5(2 − 12) in.

(900 + 30013 + 30016) ft or about 2154 ft270,000 + 90,00013

2 ft2 or about 212,942 ft2

4 − 213−2 + 213

0

1217

4113 − 316

1051x

6y + 1315y − 25 x − 3

26y15

6 313 + 4 319

−6110

12 x + 2 32x2

x25 − 222

3


Name Class Date

Simplify if possible. To start, determine if the expressions contain like radicals.

1. 315 + 415 2. 813 4 - 613 4 3. 21xy + 21y

both radicals

4. A floor tile is made up of smaller squares. Each square measures 3 in. on each side. Find the perimeter of the floor tile.

Simplify. To start, factor each radicand.

5. 118 + 132 6. 14 324 - 14 2500 7. 13 192 + 13 24

= 19 # 2 + 116 # 2

Multiply.

8. 13 - 16212 - 162 9. 15 + 15211 - 152 10. 14 + 1722

Multiply each pair of conjugates.

11. 17 - 12217 + 122 12. 11 + 313211 - 3132 13. 16 + 417216 - 4172

Practice Form K


715

712

12 − 516

47

2 314

−2 414 or −212

−415

−26

no; cannot simplify

6 313

23 + 827

−76

2412 in.


Name Class Date

Rationalize each denominator. Simplify the answer.

14. 32 + 16

15. 7 + 156 - 15

16. 1 - 21104 + 110

= 32 + 16

# 2 - 162 - 16

17. A section of mosaic tile wall has the design shown at the right. The design is made up of equilateral triangles. Each side of the large triangle is 4 in. and each side of a small triangle is 2 in. Find the total area of the design to the nearest tenth of an inch.

Simplify. Assume that all variables are positive.

18. 145 - 180 + 1245 19. 12 - 198213 + 1182 20. 62192xy2 + 423xy2

21. Error Analysis A classmate simplified the

expression 11 - 12

using the steps shown.

What mistake did your classmate make?

What is the correct answer?

22. Writing Explain the first step in simplifying 1405 + 180 - 15.



11 - 12

# 1 - 121 - 12

= 1 - 121 - 2 = 1 - 12

-1 = -1 + 12

−3 + 3216

615

47 + 131531

−36 − 1512

4 − 32110

52y13x

A ? 17.3 in.2

The student multiplied the denominator by itself instead of by its conjugate; −1 − 12

First, factor each radicand so you can combine like radicals.


Name Class Date

Multiple Choice


1. What is the simplest form of 2172 - 3132?

2172 - 3132 2412 -212 0

2. What is the simplest form of 12 - 172 11 + 2172 ?

-12 + 317 16 + 517

-12 - 317 3 + 17

3. What is the simplest form of 112 + 172112 - 172 ?

9 + 2114 9 - 2114 -5 9

4. What is the simplest form of 72 + 15

?

-14 + 715 -14 - 715

14 + 715 14 - 715

5. What is the simplest form of 813 5 - 13 40 - 213 135?

1613 5 1213 5 413 5 0

Short Response

6. A hiker drops a rock from the rim of the Grand Canyon. The distance it falls d in feet after t seconds is given by the function d = 16t2. How far has the rock fallen after (3 + 12) seconds? Show your work.

Standardized Test PrepBinomial Radical Expressions

D

F

C

F

D

[2] d = 16t2 = 16(3 + 12)2 = 16(11 + 612) = 176 + 9612 ft[1] appropriate method but with computational errors[0] incorrect answer and no work shown OR no answer given


Name Class Date

Consider how you might use a calculator to find the square of negative three. If you enter the expression -32, your calculator produces an answer of -9. However, the square of negative three is (-3)2 = (-3)(-3) = 9. Calculators follow the order of operations. Therefore, a calculator will compute -32 as the opposite of 32. The correct input is (-3)2, which is correctly evaluated as 9. Be sure to follow the order of operations when expanding binomial radical expressions.

1. Consider the algebraic expression (a + b)2. Is (a + b)2 equivalent to a2 + b2? If yes, explain. If not, explain why it is not mathematically logical and give a counterexample.

2. Are there values of a and b for which (a + b)2 = a2 + b2?

Consider each pair of expressions below for nonnegative values of the variables. State whether they are equivalent expressions. If yes, explain. If not, give a counterexample.

3. 2x2 + y2, 2x2 + 2y2

4. 1ab , 2a

b

5. 11a22, a

6. (2x2 + y2)2, x + y

EnrichmentBinomial Radical Expressions

Answer may vary. Sample: (a + b)2 means (a + b)(a + b) which, when expanded, is a2 + 2ab + b2, which is not equivalent to a2 + b2.

Answers may vary. Sample: a = 1, b = 0

Answers may vary. Sample: These expressions are not equivalent. Let x = 2 andy = 3 then 222 + 32 = 213 ≠ 24 + 29

Answers may vary. Sample: These expressions are not equivalent. Let a = 6 and

b = 2 then 162 ? 1.22 and 56

2 = 13 ? 1.73

Answers may vary. Sample: These expressions are equivalent.11a22 = 11a2 11a2 = 2a2 = a for all a # 0.

Answers may vary. Sample: These expressions are not equivalent.

(2x2 + y2)( 2x2 + y2) = 2(x2 + y2)2 = x2 + y2


Name Class Date

Two radical expressions are like radicals if they have the same index and the same radicand.

Compare radical expressions to the terms in a polynomial expression.

Like terms: 4x3 11x3 The power and the variable are the same

Unlike terms: 4y3 11x3 4y2 Either the power or the variable are not the same.

Like radicals: 413 6 1113 6 The index and the radicand are the same

Unlike radicals: 413 5 1113 6 412 6 Either the index or the radicand are not the same.

When adding or subtracting radical expressions, simplify each radical so that you can find like radicals.

Problem

What is the sum? 163 + 128

163 + 128 = 19 # 7 + 14 # 7 Factor each radicand.

= 232 # 7 + 222 # 7 Find perfect squares.

= 23227 + 22227 Use n1ab = n 1a # n1b.

= 317 + 217 Use n2an = a to simplify.

= 517 Add like radicals.

The sum is 517.

Exercises

Simplify.

1. 1150 - 124 2. 13 135 + 13 40 3. 613 - 175

4. 513 2 - 13 54 5. - 148 + 1147 - 127 6. 813 3x - 13 24x + 13 192x

ReteachingBinomial Radical Expressions

316 13

10 313x2 312

5 315

0


Name Class Date

• Conjugates,suchas1a + 1b and 1a - 1b, differ only in the sign of the second term. If a and b are rational numbers, then the product of conjugates produce a rational number:

(1a + 1b)(1a - 1b ) = (1a )2- (1b )2

= a - b.

• Youcanusetheconjugateofaradicaldenominatortorationalizethedenominator.

Problem

What is the product? (217 - 15)(217 + 15) (217 - 15)(217 + 15) These are conjugates.

= (217)2 - (15)2 Use the difference of squares formula.

= 28 - 5 = 23 Simplify.

Problem

How can you write the expression with a rationalized denominator? 4121 + 13

4121 + 13

= 4121 + 13

# 1 - 131 - 13

Use the conjugate of 1 + 13 to rationalize the denominator.

= 412 - 4161 - 3 Multiply.

= 412 - 416-2 = -

1412 - 41622 Simplify.

= -412 + 4162 = -212 + 216

Exercises

Simplify. Rationalize all denominators.

7. 13 + 16213 - 162 8. 213 + 15 - 13

9. (416 - 1)(16 + 4) 10. 2 - 17

2 + 17 11. 1218 - 62118 - 42 12. 15

2 + 13



3 13 + 12

20 + 1516

40 − 2812 215 − 115−11 + 4273


Name Class Date

Additional Vocabulary Support Rational Exponents

Choose the word or phrase from the list that best matches each sentence.

rational exponent radical form exponential form

1. The expression 24 y3 is written in .

2. A is an exponent written in fractional form.

3. The expression x 35 is written in .

Write each expression in exponential form.

4. 24 y7 =

5. (13 x)4 =

6. (15 a)3 =

7. 18 r =

Write each expression in radical form.

8. w34 =

9. b52 =

10. h12 =

11. g 37 =

Multiple Choice

12. What is 26 y413 y

in simplest terms?

y12 13 y 23 y4 y 23

radical form

rational exponent

exponential form

y 74

x 43

a35

42w3

7 2g3

2b52h

r 18

B


Name Class Date

Think About a PlanRational Exponents

Science A desktop world globe has a volume of about 1386 cubic inches. The radius of the Earth is approximately equal to the radius of the globe raised to the 10th power. Find the radius of the Earth. (Hint: Use the formula V = 4

3pr3 for the volume of a sphere.)

Know

1. The volume of the globe is e e.

2. The radius of the Earth is equal to .

Need

3. To solve the problem I need to find .

Plan

4. Write an equation relating the radius of the globe rG to the radius of the Earth rE.

5. How can you represent the radius of the globe in terms of the radius of the Earth?

6. Write an equation to represent the volume of the globe.

7. Use your previous equation and your equation from Exercise 5 to write an equation to find the radius of the Earth.

8. Solve your equation to find the radius of the Earth.

1386 in.3

the radius of the globe raised to the 10th power

the radius of the Earth

about 251,000,000 in. or 3961 mi

rE = rG10

rG = rE110

1386 = 43Pr 3

G

1386 = 43P arE

110b

3

Name Class Date


Practice Form G

Rational Exponents

Simplify each expression.

1. 12513 2. 64

12 3. 32

15

4. 712 # 7

12 5. (-5)

13 # (-5)

13 # (-5)

13 6. 3

12 # 75

12

7. 1113 # 11

13 # 11

13 8. 7

12 # 28

12 9. 8

14 # 32

14

10. 1212 # 27

12 11. 12

13 # 45

13 # 50

13 12. 18

12 # 98

12


13. x43 14. (2y)

13 15. a1.5

16. b15 17. z

23 18. (ab)

14

19. m2.4 20. t-27 21. a-1.6


22. 2x3 23. 13 m 24. 15y

25. 23 2y2 26. (14 b)3 27. 1-6

28. 2(6a)4 29. 25 n4 30. 24 (5ab)3

31. The rate of inflation i that raises the cost of an item from the present value P to

the future value F over t years is found using the formula i = (FP)

1t - 1. Round

your answers to the nearest tenth of a percent.

a. What is the rate of inflation for which a television set costing $1000 today will become one costing $1500 in 3 years?

b. What is the rate of inflation that will result in the price P doubling (that is, F = 2P) in 10 years?

5 8 2

7 −5

1411

18

4

15

4230

14.5%

7.2%

32x4

51b

52m12 171t2

151a8

32z2 41ab

322y 2a3

x 32 m

13 (5y)

12

213y

23

36a2 n45 (5ab)

34

b34 (−6)

12

Name Class Date



Rational Exponents

Write each expression in simplest form. Assume that all variables are positive.

32. (8114 ) 4 33. (32

15 ) 5 34. 125642 1

4

35. 70 36. 823 37. (-27)

23

38. x12 # x

13 39. 2y

12 # y 40. 1822 1

3

41. 3.60 42. ( 116)1

4 43. (278 )2

3

44. 28 0 45. 13x12214x

232 46.

12y13

4y12

47. (3a12 b

13 )2 48. (y

23 )-9 49. (a

23b-1

2 )-6

50. y25 # y

38 51. ax

47

x23

b 52. (2a14 )3

53. 81-12 54. (2x

25 )(6x

14 ) 55. (9x4y-2)1

2

56. a27x6

64y4b13 57.

x12y

23

x13y

12

58. y58 , y

12

59. x14 # x

16 # x

13 60. a x

-13y

x23y

-12

b 61. a 12x8

75y10b12

62. In a test kitchen, researchers have measured the radius of a ball of dough made with a new quick-acting yeast. Based on their data, the radius r of the dough ball, in centimeters, is given by r = 5(1.05)

t3 after t minutes. Round the

answers to the following questions to the nearest tenth of a cm. a. What is the radius after 5 minutes? b. What is the radius after 20 minutes? c. What is the radius after 43 minutes?

32

4

3

y16

1

x211

12x 1320

1y6

12x 76

12

2y 32

81

1

y 3140

19

9ab23

0

1

x 56

256

9

8a34

3x2y

b3

a4

94

4

5.4 cm

3x2

4y 43

x 34

y3

x22x4

5y5

x 16 y

16 y

18

6.9 cm10.1 cm


Name Class Date

Practice Form K

Rational Exponents

Simplify each expression.

1. 16 14 2. (-3)

13 # (-3)

13 # (-3)

13 3. 5

12 # 45

1214 16


4. x 14 5. x

45 6. x

29


7. 13 2 8. 23 2x2 9. 23 (2x)2

10. Bone loss for astronauts may be prevented with an apparatus that rotates

to simulate gravity. In the formula N = a0.5

2pr 0.5, N is the rate of rotation in

revolutions per second, a is the simulated acceleration in m/s2, and r is the radius of the apparatus in meters. How fast would an apparatus with the following radii have to rotate to simulate the acceleration of 9.8 m/s2 that is due to Earth’s gravity?

a. r = 1.7 m b. r = 3.6 m c. r = 5.2 m d. Reasoning Would an apparatus with radius 0.8 m need to spin faster or

slower than the one in part (a)?

2

41x

2 13

-3

0.382 rev/s0.263 rev/s0.218 rev/s

faster

52x 4

12x 22 13

15

92x 2

(2x) 23


Name Class Date


Rational Exponents

Simplify each number.

11. (-216) 13 12. 2431.2 13. 32-0.413 -216

Find each product or quotient. To start, rewrite the expression using exponents.

14. 114 62113 62 15. 25 x2

105x2 16. 120 # 13 135

= (6 14 )(6

13 )

Simplify each number.

17. (125) 23 18. (216)

23(216)

23 19. (-243)

25


20. 116x-82-34 21. 18x152- 13 22. a x2

x-10b13

23. Error Analysis Explain why the following simplification is incorrect. What is the correct simplification?

5(4 - 5 12 )

= 5(4) - 5(5 12 ) = 20 - 25

12 = 15

−6

1226 7

25

x 68

You cannot multiply 5 and

5 12 together by multiplying bases. You

have to rewrite 5 as 51 and combine the exponents; 20 − 515.

729

51x

1296

12x 5

14

6 625 5

9

x 4


Name Class Date

Standardized Test Prep Rational Exponents

Multiple Choice


1. What is 1213 # 45

13 # 50

13 in simplest form?

127,000 30 10713 27,000

2. What is x13 # y


x32y3 2xy3 23 (xy)2 23 xy2

3. What is x13 # x

12 # x


x 1312 x 1

24 x 19 x 524

4. What is £ x23y

13

x12y

34≥6

in simplest form?

xy 52 x 7y 52 1

xy 52

x

y 52

5. What is (-32x10 y35)-15 in simplest form?

2x2y7 - 2x2y7 - 1

2x2 y7 2x2 y7

Short Response

6. The surface area S, in square units, of a sphere with volume V, in cubic units, is given by the formula S = p 13(6V )

23. What is the surface area of a sphere with

volume 43 mi3? Show your work.

[2] S = P13(6V)

23 = P

13 c6 a4

3bd23 = P

13(8)

23 = 4P

13mi2

[1] appropriate method but some computational errors[0] incorrect answer and no work shown OR no answer given

B

I

A

I

C


Name Class Date

Enrichment Rational Exponents

Power Games

Each problem below involves rational exponents. Some of the problems are tricky. Good luck!

1. Begin with any positive number. Call it x. Divide x by 2. Call the result r. Now follow these directions carefully. You may use a calculator.

a. Divide x by r. Call the result q. b. Add q and r. Call the result s. c. Divide s by 2. Call the result r. d. Go back to step a.

Repeat steps a –d until r no longer changes. What is the relationship between the original x and the final result?

2. If we take the square root of a number 6 times, it would look like this:

654321x

Rewrite the expression above using rational exponents.

Simplify the expression above. Express the denominator of the exponent as a

power of 2.

If you were to take the square root of a number 10 times, what would the denominator of the exponent be equal to if you use rational exponents? 12 times?

Choose any number and repeatedly take the square root. What number is the answer approaching?

Does the answer appear to approach the same number if you change the number you choose?

In Exercises 3–6, assume that the square roots and the operations inside them repeat forever.

3. How much is 42 * 32 * 22 * 12 * c ? (Hint: Let

y = 42 * 32 * 22 * 12 * c . Then use substitution and solve the

equation y = 22 * y.)

4. How much is 42 + 32 + 22 + 12 + c ?

5. How much is 42 - 32 - 22 - 12 - c ?

6. How much is 42 , 32 , 22 , 12 , c ?

final r = 1x

aaaaax12b

12b

12b

12b

12b

12

x164; 26

210; 212

1

yes

2

2

1

322


Name Class Date

Reteaching Rational Exponents

You can simplify a number with a rational exponent by converting the expression to a radical expression:

x1n =

1nx, for n 7 0 9

12 = 219 = 3 8

13 = 13 8 = 2

You can simplify the product of numbers with rational exponents m and n by raising the number to the sum of the exponents using the rule

am # an = am+n

Problem

What is the simplified form of each expression?

a. 3614 # 36

14

3614 # 36

14 = 36

14+1

4 Use am # an = am+n.

= 3612 Add.

= 2136 Use x

1n = n1x.

= 6 Simplify.

b. Write 16x23212x

342 in simplified form.

16x23212x

342 = 6 # 2 # x

23 # x

34 Commutative and Associative

Properties of Multiplication

= 6 # 2 # x23+3

4 Use xm # xn = xm+n.

= 12x1712 Simplify.

Exercises

Simplify each expression. Assume that all variables are positive.

1. 513 # 5

23 2. 12y

14213y

132 3. (-11)

13 # (-11)

13 # (-11)

13

4. -y23 y

15 5. 5

14 # 5

14 6. 1 -3x 16217x 262

5 6y 712 −11

−211x15−y 1315


Name Class Date


Rational Exponents

To write an expression with rational exponents in simplest form, simplify all exponents and write every exponent as a positive number using the following rules for a ≠ 0 and rational numbers m and n:

a-n = 1an 1

a-m = am (am)n = amn (ab)m = ambm

Problem

What is 18x9y-32-23 in simplest form?

(8x9y-3)-23 = 123 x9 y-32-2

3 Factor any numerical coefficients.

= 1232-23 1x92-2

3 1y-32-23 Use the property (ab)m = ambm.

= 2-2x-6y2 Multiply exponents, using the property (am)n = amn.

= y2

22x6 Write every exponent as a positive number.

= y2

4x6 Simplify.

Exercises


7. 116x2 y82-12 8. 1z-32 1

9 9. 12x1424

10. 125x-6 y22 12 11. 18a-3 b92 2

3 12. a16z4

25x8b-1

2

13. a x2

y-1b15 14. 127m9 n-32-2

3 15. a32r2

2s4 b14

16. 19z102 32 17. (-243)-1

5 18. ax25

y12b

10

14xy4 16x

2r 12s

5yx3

4b6

a25x4

4z2

−13

n2

9m6

x4

y527z15

1

z 13

x25y

15


Name Class Date

Additional Vocabulary SupportSolving Square Root and Other Radical Equations

Problem

Solve the equation 423 (y + 2)2 + 3 = 19. Justify your steps. Then check your solution.

4(y + 2)23 + 3 = 19 Rewrite the radical using a rational exponent.

4(y + 2)23 = 16 Subtract 3 from each side.

(y + 2)23 = 4 Divide each side by 4.

c (y + 2)23 d

32

= 432 Raise each side to the 32 power.

(y + 2) = 8 Simplify.

y = 6 Solve for y.

Check 4 23 (6 + 2)2 + 3 ≟ 19 Substitute 6 for y.

423 82 + 3 ≟ 19 Add.

4 # 4 + 3 ≟ 19 Simplify the radical.

19 = 19 Simplify.

Exercise

Solve the equation 92(2x - 4)4 + 2 = 38. Justify your steps. Then check your solution.

9(2x - 4)42 + 2 = 38

9(2x - 4)2 = 36

(2x - 4)2 = 4

c (2x - 4)2 d12

= 412

(2x - 4) = 2

x = 3

Check 92(2 # 3 - 4)4 + 2 ≟ 38

9216 + 2 ≟ 38

38 = 38

Rewrite the radical using a rational exponent.

Divide each side by 9.

Raise each side to the 12 power.

Simplify.

Solve for x.

Substitute 3 for x.

Simplify the expression under the radical.

Simplify.

Subtract 2 from each side and simplify the exponent.


Name Class Date

Think About a PlanSolving Square Root and Other Radical Equations

Traffic Signs A stop sign is a regular octagon, formed by cutting triangles off the corners of a square. If a stop sign measures 36 in. from top to bottom, what is the length of each side?

Understanding the Problem

1. How can you use the diagram at the right to find a relationship between s and x?

.

2. How can you use the diagram at the right to find another relationship between s and x?

.

3. What is the problem asking you to determine?

Planning the Solution

4. What are two equations that relate s and x?

5. How can you use your equations to find s?

.

Getting an Answer

6. Solve your equations for s.

7. Is your answer reasonable? Explain.

.

x

x

x

s

s

36 in.

Since the triangles are right triangles, use the

Pythagorean Theorem to relate s and x

The length of a side of the square, which is s 1 2x, is the same as the height of the

stop sign from top to bottom

Yes; the length of one side of the stop sign is a little more than a third of the total

height of the sign

the length s of each side of the stop sign

Solve the first equation for x and substitute the result into the second equation

2x2 ∙ s2; 2x ∙ s ∙ 36

about 14.9 in.

Name Class Date


Practice Form G

Solving Square Root and Other Radical Equations

Solve.

1. 51x + 2 = 12 2. 31x - 8 = 7 3. 14x + 2 = 8

4. 12x - 5 = 7 5. 13x - 3 - 6 = 0 6. 15 - 2x + 5 = 12

7. 13x - 2 - 7 = 0 8. 14x + 3 + 2 = 5 9. 133 - 3x = 3

10. 13 2x + 1 = 3 11. 13 13x - 1 - 4 = 0 12. 13 2x - 4 = -2

Solve.

13. (x - 2) 13 = 5 14. (2x + 1)13 = -3 15. 2x 34 = 16

16. 2x 13 - 2 = 0 17. x 12 - 5 = 0 18. 4x 32 - 5 = 103

19. (7x - 3)12 = 5 20. 4x

12 - 5 = 27 21. x 16 - 2 = 0

22. (2x + 1)13 = 1 23. (x - 2)

23 - 4 = 5 24. 3x 43 + 5 = 53

25. The formula P = 41A relates the perimeter P, in units, of a square to its area A, in square units. What is the area of the square window shown below?

26. The formula A = 6V 23 relates the surface area A, in square units, of a cube to

the volume V, in cubic units. What is the volume of a cube with surface area 486 in.2?

27. A mound of sand at a rock-crushing plant is growing over time. Th e equation t = 13 5V - 1 gives the time t, in hours, at which the mound has volume V, in cubic meters. When is the volume equal to 549 m3?

Perimeter: 24 ft

4

27

17

13

127

1

4

0

36 ft2

729 in.3

14 h

29, 225 8, 28

64 64

25 9

214 16

5 22

32

8

13 222

25 9

Name Class Date


28. City officials conclude they should budget s million dollars for a new library building if the population increases by p thousand people in a ten-year census. The formula s = 2 + 13(p + 1)

25 expresses the relationship between population and

library budget for the city. How much can the population increase without the city going over budget if they have $5 million for a new library building?

Solve. Check for extraneous solutions.

29. 1x + 1 = x - 1 30. 12x + 1 = -3

31. (x + 7)12 = x - 5 32. (2x - 4)

12 = x - 2

33. 1x + 2 = x - 18 34. 1x + 6 = x

35. (2x + 1)12 = -5 36. (x + 2)

12 = 10 - x

37. 1x + 1 = x + 1 38. 19 - 3x = 3 - x

39. 13 2x - 4 = -2 40. 215 5x + 2 - 1 = 3

41. 14x + 2 = 13x + 4 42. 17x - 6 - 15x + 2 = 0

43. 2(x - 1)12 = (26 + x)

12 44. (x - 1)

12 - (2x + 1)

14 = 0

45. 12x - 1x + 1 = 1 46. 17x - 1 = 15x + 5

47. (7 - x)12 = (2x + 13)

12 48. (x - 7)

12 = (x + 5)

14

49. 1x + 9 - 1x = 1 50. 13 8x - 13 6x - 2 = 0

51. A clothing manufacturer uses the model a = 1f + 4 - 136 - f to estimate the amount of fabric to order from a mill. In the formula, a is the number of apparel items (in hundreds) and f is the number of units of fabric needed. If 400 apparel items will be manufactured, how many units of fabric should be ordered?

52. What are the lengths of the sides of the trapezoid shown at the right if the perimeter of the trapezoid is 17 cm?



x � 1

x

2!x2!x

242,000

3

9

23

no solution

21, 0

22

2

10

8

22

16

32

x ∙ 4 cm, 21x ∙ 4 cm, x ∙ 1 ∙ 5 cm

21

11

3

4

4

6

0, 3

7

no solution

2, 4

9

Name Class Date


Solve. To start, rewrite the equation to isolate the radical.

1. 1x + 2 - 2 = 0 2. 12x + 3 - 7 = 0 3. 2 + 13x - 2 = 61x + 2 = 2

Solve.

4. 2(x - 2) 23 = 50 5. 2(x + 3)

32 = 54 6. (6x - 5)

13 + 3 = -2

7. The formula d = 22 Vph

relates the diameter d, in units, of a cylinder to its volume V, in cubic units, and its height h, in units. A cylindrical can has a diameter of 3 in. and a height of 4 in. What is the volume of the can to the nearest cubic inch?

8. Writing Explain the difference between a radical equation and a polynomial equation.

9. Reasoning If you are solving 4(x + 3) 34 = 7, do you need to use the absolute

value to solve for x? Why or why not?

Practice Form K


223

127 and ∙123 6 ∙20

28 in.3

6

A radical equation has a variable in a radicand or a variable with afractional exponent, while a polynomial equation has a variable with whole number exponents.

No; the numerator of the exponent 34 is not even.

Name Class Date




Solve. Check for extraneous solutions. First, isolate a radical, then square each side of the equation.

10. 14x + 5 = x + 2 11. 1-3x - 5 - 3 = x 12. 1x + 7 + 5 = x

114x + 522 = (x + 2)2

13. 12x - 7 = 1x + 2 14. 13x + 2 - 12x + 7 = 0 15. 12x + 4 - 2 = 1x

112x - 722 = 11x + 222

16. Find the solutions of 1x + 2 = x. a. Are there any extraneous solutions? b. Reasoning How do you know the answer to part (a)?

17. A floor is made up of hexagon-shaped tiles. Each hexagon tile has an area of 1497 cm2. What is the length of each side of the hexagon? (Hint: Six equilateral triangles make one hexagon.)

s

s!32

1 and ∙1

∙2 9

9

5 0 and 16

2∙1

Substitute the solutionsinto the original equation. If a solution does not make the equation true, then the solution is extraneous.

about 24 cm


Name Class Date

Gridded Response

Solve each exercise and enter your answer in the grid provided.

1. What is the solution? 12x - 4 - 3 = 1

2. What is the solution? 5x 12 - 8 = 7

3. What is the solution? 12x - 6 = 3 - x

4. What is the solution? 15x - 3 = 12x + 3

5. Kepler’s Third Law of Orbital Motion states that the period P (in Earth years) it takes a planet to complete one orbit of the sun is a function of the distance d (in astronomical units, AU) from the planet to the sun. This relationship is P = d

32. If it takes Neptune 165 years to orbit the sun, what is the distance

(in AU) of Neptune from the sun? Round your answer to two decimal places.

Standardized Test Prep Solving Square Root and Other Radical Equations

1. 2. 3. 4. 5.

Answers

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

−

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

−

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

−

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

−

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

−1 0

10

9

9

3

3

22

2

3 0 . 0 8

3

0 0

8


Name Class Date

When solving radical equations you will often get an extraneous solution. You can use a graph to explain why an algebraic answer is not a solution.

1. Solve the equation 1x + 2 = x - 4. Is there an extraneous solution?

2. To analyze this equation with a graph, rewrite the equation as a system of two equations. What two equations can you write?

3. Graph the two equations.

4. Explain how you find the solution to this system of equations on your graph. What is the solution?

5. How can you use the solution from the graph of the system of equations to help you solve the original equation 1x + 2 = x - 4?

6. How can you tell from your graph that one of your algebraic answers is an extraneous solution?

Solve each equation. Graph each equation as a system to determine if there are any extraneous solutions.

7. 14x + 1 = 3 8. x = 16 - x 9. 1x + 1 = x - 1

Enrichment Solving Square Root and Other Radical Equations

7; 2 is an extraneous solution.

y ∙ 1x ∙ 2 and y ∙ x ∙ 4

Answers may vary. Sample: On a graph the solution to a system of equations is the point of intersection; the solution for this system is (7, 3).

Answers may vary. Sample: The x-coordinate of the solution to the system is the solution to the original equation.

Answers may vary. Sample: Because there is only one point of intersection, there can only be one solution to the equation.

2; no extraneous solutions

2; ∙3 is an extraneous solution.

3; 0 is an extraneous solution.

4 8

�8

�4

4

8

x

y

O�8 �4

4 8

�8

�4

4

8 y

xO�8 �4 �4 84

�8

�4

4

8

x

y

�4 4 8

�8

�4

4

8

x

y


Name Class Date

Reteaching Solving Square Root and Other Radical Equations

Equations containing radicals can be solved by isolating the radical on one side of the equation, and then raising both sides to the same power that would undo the radical.

Problem

What is the solution of the radical equation? 212x + 2 - 2 = 10

212x + 2 - 2 = 10

212x + 2 = 12 Add 2 to each side.

12x + 2 = 6 Divide each side by 2.

(12x + 2)2 = 62 Square each side to undo the radical.

2x + 2 = 36 Simplify.

2x = 34 Subtract 2 from each side.

x = 17 Divide each side by 2.

Check the solution in the original equation.

Check

212x + 2 - 2 = 10 Write the original equation.

212(17) + 2 - 2 ≟ 10 Replace x by 17.

2136 - 2 ≟ 10 Simplify.

12 - 2 ≟ 10

10 = 10

The solution is 17.

Exercises

Solve. Check your solutions.

1. x 12 = 13 2. 312x = 12 3. 13x + 5 = 11

4. (3x + 4)12 - 1 = 4 5. (6 - x) 12 + 2 = 5 6. 13x + 13 = 4

7. (x + 2) 12 - 5 = 0 8. 13 - 2x - 2 = 3 9. 13 5x + 2 - 3 = 0

169

7 ∙ 3 1

∙ 11 523

8 12


Name Class Date

Reteaching (continued) Solving Square Root and Other Radical Equations

An extraneous solution may satisfy equations in your work, but it does not make the original equation true. Always check possible solutions in the original equation.

Problem

What is the solution? Check your results. 117 - x - 3 = x

117 - x - 3 = x

117 - x = x + 3 Add 3 to each side to get the radical alone on one side of the equal sign.

1117 - x22 = (x + 3)2 Square each side.

17 - x = x2 + 6x + 9

0 = x2 + 7x - 8 Rewrite in standard form.

0 = (x - 1)(x + 8) Factor.

x - 1 = 0 or x + 8 = 0 Set each factor equal to 0 using the Zero Product Property.

x = 1 or x = -8Check

117 - x - 3 = x 117 - x - 3 = x

117 - 1 - 3 ≟ 1 117 - (-8) - 3 ≟ -8

116 - 3 ≟ 1 125 - 3 ≟ -8

1 = 1 2 ≠ -8

The only solution is 1.

Exercises

Solve. Check for extraneous solutions.

10. 15x + 1 = 14x + 3 11. 1x2 + 3 = x + 1 12. 13x = 1x + 6

13. x = 1x + 7 + 5 14. x - 31x - 4 = 0 15. 1x + 2 = x - 4

16. 12x - 10 = x - 5 17. 13x - 6 = 2 - x 18. 1x - 1 + 7 = x

19. 15x + 1 = 13x + 15 20. 1x + 9 = x + 7 21. x - 1x + 2 = 40

5, 7 2

2

9 16 7

3solutionno

7 ∙ 5

10

47


Name Class Date

Additional Vocabulary Support Function Operations

Darnell wrote the steps to compose the following functions on index cards, but the cards got mixed up.

Let f (x) = x + 7 and g (x) = x3. What is (g ∘ f )( -4)?

Use the note cards to write the steps in order.

1. First,

.

2. Second,

.

3. Then,

.

4. Finally,

.

Subtract 4 from 7.

Raise 3 to the 3rd power.

Substitute -4 for x in f(x).

Substitute 3 into g(x).

substitute ∙4 for x in f(x)

subtract 4 from 7

substitute 3 into g(x)

raise 3 to the 3rd power


Name Class Date

Think About a Plan Function Operations

Sales A salesperson earns a 3% bonus on weekly sales over $5000. Consider the following functions.

g(x) = 0.03x h(x) = x - 5000

a. Explain what each function above represents.

b. Which composition, (h ∘ g)(x) or (g ∘ h)(x), represents the weekly bonus? Explain.

1. What does x represent in the function g(x)?

2. What does the function g(x) represent?

3. What does x represent in the function h(x)?

4. What does the function h(x) represent?

5. What is the meaning of (h ∘ g)(x)?

.

6. Assume that x is $7000. What is (h ∘ g)(x)?

7. What is the meaning of (g ∘ h)(x)?

.

8. Assume that x is $7000. What is (g ∘ h)(x)?

9. Which composition represents the weekly bonus? Explain

.

the sales amount used to calculate a 3% bonus

the bonus earned by the salesperson on sales

the total weekly sales made by the salesperson

the weekly sales over $5000 made by the salesperson

First multiply the value of x by 0.03, then subtract 5000 from the result

(g ∘ h)(x) represents the weekly bonus because you must first find the sales amount

over 5000 by subtracting 5000 from the weekly sales, and then you multiply the result

by the bonus percent as a decimal, or 0.03

First subtract 5000 from the value of x, then multiply the result by 0.03

−4790

60


Name Class Date

Practice Form G

Function Operations

Let f (x) = 4x - 1 and g(x) ∙ 2x2 ∙ 3. Perform each function operation and then find the domain.

1. (f + g)(x) 2. (f - g)(x) 3. (g - f )(x)

4. (f # g)(x) 5. fg (x) 6.

gf (x)

Let f (x) ∙ 2x and g (x) ∙ 1x ∙ 1. Perform each function operation and then find the domain of the result.

7. (f + g)(x) 8. (f - g)(x) 9. (g - f )(x)

10. ( f # g)(x) 11. fg (x) 12.

gf (x)

Let f (x) ∙ ∙3x ∙ 2, g (x) ∙ x5, h(x) ∙∙2x2 ∙ 9, and j(x) ∙ 5 ∙ x. Find each value or expression.

13. ( f ∘ j)(3) 14. ( j ∘ h)(-1) 15. (h ∘ g)(-5)

16. (g ∘ f )(a) 17. ƒ(x) + j(x) 18. ƒ(x) - h(x)

19. (g ∘ f )(-5) 20. ( f ∘ g)(-2) 21. 3ƒ(x) + 5g(x)

22. g( f (2)) 23. g( f (x)) 24. f (g(1))

25. A video game store adds a 25% markup on each of the games that it sells. In addition to the manufacturer’s cost, the store also pays a $1.50 shipping charge on each game.

a. Write a function to represent the price f (x) per video game after the store’s markup.

b. Write a function g(x) to represent the manufacturer’s cost plus the shipping charge.

c. Suppose the manufacturer’s cost for a video game is $13. Use a composite function to find the cost at the store if the markup is applied after the shipping charge is added.

d. Suppose the manufacturer’s cost for a video game is $13. Use a composite function to find the cost at the store if the markup is applied before the shipping charge is added.

2x2 ∙ 4x ∙ 2; all real numbers

8x3 ∙ 2x2 ∙ 12x ∙ 3; all real numbers

2x ∙ 2x ∙ 1; x # 0

2x2x ∙ 2x; x # 02x1x ∙ 1

; x # 0 and x ≠ 1

2x ∙ 2x ∙ 1; x # 0 ∙2x ∙ 2x ∙ 1; x # 0

1x ∙ 12x ; x + 0

∙2x2 ∙ 4x∙4; all real numbers

4x ∙ 12x2 ∙ 3

; all real numbers 2x2 ∙ 34x ∙ 1 ; all real numbers except 14

2x2 ∙ 4x ∙ 4 all real numbers

∙4

175

∙ 45 ∙3x ∙ 25

165

∙3a ∙ 25

∙2

∙ 4x ∙ 7 2x2 ∙ 3x ∙ 7

∙8x ∙ 6

75

7

f(x) ∙ 1.25x

g(x) ∙ x ∙ 1.5

f(g(13)) ? $18.13

g(f(13)) ∙ $17.75


Name Class Date

26. The formula V = 43pr3 expresses the relationship between the volume V and

radius r of a sphere. A weather balloon is being inflated so that the radius is changing with respect to time according to the equation r = t + 1, where t is the time, in minutes, and r is the radius, in feet.

a. Write a composite function f (t) to represent the volume of the weather balloon after t minutes. Do not expand the expression.

b. Find the volume of the balloon after 5 minutes. Round the answer to two decimal places. Use 3.14 for π.

27. A boutique prices merchandise by adding 80% to its cost. It later decreases by 25% the price of items that do not sell quickly.

a. Write a function f (x) to represent the price after the 80% markup. b. Write a function g(x) to represent the price after the 25% markdown. c. Use a composition function to find the price of an item, after both price

adjustments, that originally costs the boutique $150. d. Does the order in which the adjustments are applied make a difference?

Explain.

28. A department store has marked down its merchandise by 25%. It later decreases by $5 the price of items that have not sold.

a. Write a function f (x) to represent the price after the 25% markdown. b. Write a function g(x) to represent the price after the $5 markdown. c. Use a composition function to find the price of a $50 item after both price

adjustments. d. Does the order in which the adjustments are applied make a difference?

Explain.

Let g(x) ∙ x2 ∙ 5 and h(x) ∙ 3x ∙ 2. Perform each function operation.

29. (h ∘ g)(x) 30. g(x) # h(x) 31. -2g(x) + h(x)


Function Operations

f(t) ∙ 43 P(t ∙ 1)3

904.32 ft3

f(x) ∙ 1.8x

f(x) ∙ 0.75xg(x) ∙ x ∙ 5

g(x) ∙ 0.75x

g(f (150)) ∙ $202.50

No; it doesn’t matter whether you first multiply by 0.75 or by 1.8.

Yes; multiplying by 0.75 and then subtracting by 5 is different than subtracting by 5 and then multiplying by 0.75.

g(f (50)) ∙ $32.50

3x2 ∙ 13 3x3 ∙ 2x2 ∙ 15x ∙ 10 ∙2x2 ∙ 3x ∙ 12


Name Class Date

Practice Form K

Function Operations

Let f (x) ∙ 4x ∙ 8 and g(x) ∙ 2x ∙ 12. Perform each function operation and then find the domain of the result.

1. ( f + g)(x) 2. ( f - g)(x) 3. ( f # g)(x) 4. a fgb(x)

f (x) + g(x)

Let f (x) ∙ x ∙ 2 and g(x) ∙ 1x ∙ 1. Perform each function operation and then find the domain of the result.

5. ( f + g)(x) 6. ( f # g)(x) 7. a fgb(x) 8. ag

f b(x)

Let f (x) ∙ x ∙ 2 and g(x) ∙ x2. Find each value. To start, use the definition of composing functions to find a function rule.

9. (g 5 f )(4) 10. ( f 5 g)(-1) 11. (g 5 f )(-3)

f (4) = 4 - 2 = 2

Let f (x) ∙ 1x and g(x) ∙ (x ∙ 2)2. Find each value.

12. ( f 5 g)(-5) 13. ( f 5 g)(0) 14. (g 5 f )(4)

(f ∙ g) (x) ∙ 6x ∙ 4; all real numbers

(f ∙ g)(x)∙ x ∙ 1x ∙ 1;all x # 0

(f # g)(x)∙ x1x ∙ x ∙ 21x ∙ 2;all x # 0

( fg)(x) ∙ x ∙ 21x ∙ 1

;

all x # 0, x ≠ 1

(gf )(x) ∙ 1x ∙ 1

x ∙ 2 ;

all x # 0

(f ∙ g) (x)∙ 2x ∙ 20; all real numbers

(f # g) (x)∙ 8x2 ∙ 32x ∙ 96; all real numbers

( fg)(x) ∙ 2x ∙ 4

x ∙ 6 ;

all real numbers, x ≠ 6

4∙1 25

3 2 16


Name Class Date


Function Operations

15. A car dealer offers a 15% discount off the list price x of any car on the lot. At the same time, the manufacturer offers a $1000 rebate for each purchase of a car.

a. Write a function f (x) to represent the price after discount. b. Write a function g (x) to represent the price after the $1000 rebate. c. Suppose the list price of a car is $18,000. Use a composite function to find the price

of the car if the discount is applied before the rebate. d. Suppose the list price of a car is $18,000. Use a composite function to find the price

of the car if the discount is applied after the rebate. e. Reasoning Between parts (c) and (d), will the dealer want to apply the

discount before or after the rebate? Why?

16. Error Analysis f (x) = 21x and g(x) = 3x - 6. Your friend gives a domain

for a fgb(x) as x Ú 0. Is this correct? If not, what is the correct domain?

Let f (x) ∙ 2x2 ∙ 3 and g(x) ∙ x ∙ 12 . Find each value.

17. f (g(2)) 18. g( f (-3)) 19. ( f 5 f )(-1)

20. Reasoning A local bookstore has a sale on all their paperbacks giving a 10% discount. You also received a coupon in the mail for $4 off your purchase. If you buy 2 paperbacks at $8 each, is it less expensive for you to apply the discount before the coupon or after the coupon? How much will you save? before the coupon; $.40

f(x) ∙ 0.85xg(x) ∙ x ∙ 1000

$14,300

$14,450

After; they will make more money selling the car for a higher price.

No; the correct domain is x # 0, x ≠ 2.

32

8 ∙1


Name Class Date

Multiple Choice

For Exercises 1∙5, choose the correct letter.

1. Let f (x) = - 2x + 5 and g(x) = x3. What is (g - f )(x)?

x3 - 2x + 5 - x3 - 2x + 5

x3 + 2x - 5 - x3 + 2x - 5

2. Let f (x) = 3x and g(x) = x2 + 1. What is ( f # g)(x)?

9x2 + 3x 9x2 + 1 3x3 + 3x 3x3 + 1

3. Let f (x) = x2 - 2x - 15 and g(x) = x + 3. What is the domain of fg (x)?

all real numbers x ≠ - 3

x ≠ 5, - 3 x 7 0

4. Let f (x) = 1x + 1 and g(x) = 2x + 1. What is (g ∘ f )(x)?

21x + 3 12x + 1 + 1

2x1x + 2x + 1x + 1 2x + 1x + 2

5. Let f (x) = 1x and g(x) = x2 - 2. What is ( f ∘ g)( - 3)?

179 1

7 - 179 - 73

Short Response

6. Suppose the function f (x) = 0.035x represents the number of U.S. dollars equivalent to x Russian rubles and the function g(x) = 90x represents the number of Japanese yen equivalent to x U.S. dollars. Write a composite function that represents the number of Japanese yen equivalent to x Russian rubles. Show your work.

Standardized Test PrepFunction Operations

[2] (g 5 f )(x) ∙ g(f(x)) ∙ 90(0.035x) ∙ 3.15x

B

H

C

F

B

[1] appropriate method but with one computational error

[0] incorrect answer and no work shown OR no answer given


Name Class Date

Composition and Linear FunctionsTwo functions f (x) and g(x) are equal if they have the same domains and the same value for each point in their domain. Suppose that f (x) = Ax + B and g(x) = Cx + D are two linear functions both of whose domains are the set of real numbers.

1. If f (x) = g(x), what can you conclude by examining the values of f and g at x = 0?

2. Use your conclusion to eliminate D from the definition of g(x).

3. What equation results from examining the values of f and g at x = 1?

4. What can you conclude about A and C?

5. When are two linear functions equal?

6. Compute (g ∘ f )(x).

7. What type of function is the composite of two linear functions?

8. What is the coefficient of x in the expression for (g ∘ f )(x)?

9. What is the constant term?

10. Compute ( f ∘ g)(x) and express it in slope-intercept form.

11. What equation must be satisfied if f ∘ g = g ∘ f ?

12. What equations must be satisfied if f ∘ g = f ?

13. What equations must be satisfied if f ∘ g = g?

14. What must occur if f ∘ g = 0?

15. Constant functions are a subset of linear functions in which the coefficient of x is zero. What type of function is the composite of two constant functions?

16. a. If h(x) and k(x) are two constant functions, under what circumstances does h ∘ k = k ∘ h?

b. Under what circumstances does h ∘ k = k? c. Under what circumstances does h ∘ k = h? d. Under what circumstances does h ∘ k = 0?

EnrichmentFunction Operations

g(x) ∙ Cx ∙ B

A ∙ B ∙ C ∙ B

A ∙ C

ACx ∙ BC ∙ D

AC

BC ∙ D

ACx ∙ AD ∙ B

BC ∙ D ∙ AD ∙ B

AC ∙ A and AD ∙ B ∙ B

AC ∙ C and AD ∙ B ∙ D

AC ∙ 0 and AD ∙ B ∙ 0

constant

if h(x) ∙ k(x)if h(x) ∙ k(x)h 5 k always equals hif h(x) ∙ 0

when the coefficients of x are equal and the constant terms are equal

B ∙ D

linear


Name Class Date

When you combine functions using addition, subtraction, multiplication, or division, the domain of the resulting function has to include the domains of both of the original functions.

Problem

Let f (x) = x2 - 4 and g(x) = 1x. What is the solution of each function operation? What is the domain of the result?

a. ( f + g)(x) = f (x) + g(x) = (x2 - 4) + (1x) = x2 + 1x - 4

b. ( f - g)(x) = f (x) - g(x) = (x2 - 4) - (1x) = x2 - 1x - 4

c. (g - f )(x) = g(x) - f (x) = (1x) - (x2 - 4) = -x2 + 1x + 4

d. ( f # g)(x) = f (x) # g(x) = (x2 - 4)(1x) = x21x - 41x

The domain of f is all real numbers. The domain of g is all x Ú 0. For parts a–d, there are no additional restrictions on the values for x, so the domain for each of these is x Ú 0.

e. fg (x) = f (x)

g(x) = x2 - 41x= (x2 - 4)1x

x

As before, the domain is x Ú 0. But, because the denominator cannot be zero, eliminate any values of x for which g(x) = 0. The only value for which 1x = 0 is x = 0. Therefore,

the domain of fg is x 7 0.

f. gf (x) = g(x)

f(x) = 1xx2 - 4

Similarly, begin with x Ú 0 and eliminate any values of x that make the denominator f (x) zero: x2 - 4 = 0 when x = - 2 and x = 2. Therefore, the domain of

gf is x Ú 0

combined with x ≠ -2 and x ≠ 2. In other words, the domain is x Ú 0 and x ≠ 2, or all

nonnegative numbers except 2.

Exercises

Let f (x) ∙ 4x ∙ 3 and g(x) ∙ x2 ∙ 2. Perform each function operation and then find the domain of the result.

1. ( f + g)(x) 2. ( f - g)(x) 3. (g - f )(x)

4. ( f # g)(x) 5. fg (x) 6.

gf (x)

ReteachingFunction Operations

x2 ∙ 4x ∙ 1; all real numbers

4x3 ∙ 3x2 ∙ 8x ∙ 6; all real numbers

∙ x2 ∙ 4x ∙ 5; all real numbers

4x ∙ 3x2 ∙ 2

; all real numbers

x2 ∙ 4x ∙ 5; all real numbers

x2 ∙ 24x ∙ 3; x ≠ 3

4


Name Class Date

• Onewaytocombinetwofunctionsisbyformingacomposite.

• Acompositeiswritten(g ∘ f ) or g( f (x)). The two different functions are g and f.

• Evaluatetheinnerfunctionf (x) first.

• Usethisvalue,thefirstoutput,astheinputforthesecondfunction,g(x).

Problem

What is the value of the expression g( f (2)) given the inner function, f (x) = 3x - 5 and the outer function, g(x) = x2 + 2?

Exercises

Evaluate the expression g( f (5)) using the same functions for g and f as in the Example. Fill in blanks 7–14 on the chart.

Use one color highlighter to highlight the first input. Use a second color to highlight the first output and the second input. Use a third color to highlight the second output, which is the answer.

Given f (x) ∙ x2 ∙ 4x and g(x) ∙ 2x ∙ 3, evaluate each expression.

15. f (g(2)) 16. g( f (2.5)) 17. g( f ( - 5)) 18. f (g( - 5))


Function Operations

1st inputx � 2

f(x)

32nd output

11st output

1st output, 1,becomes 2nd input

2nd output

3

3x � 5 x2 � 2

g(x)

3(2) � 5

6 � 5

12 � 2

1 � 2

1st inputx � 5

3x � 5

3(—) � 5

— � 5

———

f(x)

x2 � 2

—2

� 2

———

g(x)1st output,

———,becomes 2nd input

2nd output

———

7.

8.

9.

11.

12.

13.

10. 14.

— � 2

77 35.5 13 21

5

15

10

10

10

100

102

102


Name Class Date

Additional Vocabulary Support Inverse Relations and Functions

Choose the word or phrase from the list that best matches each sentence.

inverse relation inverse function one-to-one function f−1

1. In a , each y-value in the range corresponds to exactly one x-value in the domain.

2. A relation pairs element a of its domain to element b of its range. The pairs b with a.

3. The inverse of a function f is represented by .

4. If a relation and its inverse are functions, then they are .

5. Circle the inverse of y = 2x + 1. y = x - 12 x = y

2 - 2 y = x - 12

6. Circle the inverse of y = (3 - x)2. y = 3 + 1x y = 3 - 1x y = 9 - 1x

7. Circle the inverse of y = 5x2 + 4. y = x - 45 y = 2x - 4

5 y = 2x - 45

8. Explain each of the steps followed to find f -1 of f (x) = 13x - 2.

y = 13x - 2

x = 13y - 2

x2 = 3y - 2

x2 + 2 = 3y

y = x2 + 23

9. Find f -1 for f(x) = 4x - 8, and explain the steps.

y = 4x - 8

x = 4y - 8

x + 8 = 4y

y = x4 + 2

one-to-one function

inverse relation

inverse functions

Replace f (x) with y.

Switch x and y.

Square both sides.

Add 2 to both sides.

Replace f (x) with y.

Switch x and y.

Add 8 to both sides.

Divide both sides by 4 and solve for y.

Divide both sides by 3 and solve for y.

f−1


Name Class Date

Think About a Plan Inverse Relations and Functions

Geometry Write a function that gives the length of the hypotenuse of an isosceles right triangle with side length s. Evaluate the inverse of the function to find the side length of an isosceles right triangle with a hypotenuse of 6 in.

Know

1. An equation that relates the length of each side s and the length of the hypotenuse h of an isosceles right triangle is

e e.

Need

2. To solve the problem I need to:

.

Plan

3. A function that gives the length of the hypotenuse h in terms of the side length

s is e e.

4. An inverse function that gives the side length s in terms of the length of the

hypotenuse h is

.

5. What is the value of the inverse function for h = 6 in.?

6. Is the side length reasonable? Explain.

.

s

sh

2s2 = h2

write a function for the length of the hypotenuse in terms

of the side length and then find the inverse function for

the side length in terms of the hypotenuse

Yes; the side length is less than the length of the hypotenuse

h = s22

s = h12

about 4.24 in.


Name Class Date

Practice Form G

Inverse Relations and Functions

Find the inverse of each relation. Graph the given relation and its inverse.

1.

y −3 −2 −1

x −2 −1 0 1

0

2.

y −3 −1 0

x 0 1 2 3

−2

3.

y −1 0 1

x −3 −1 1 2

3

4.

y 3 2 1

x −3 −2 −1 0

0

Find the inverse of each function. Is the inverse a function?

5. y = x2 + 2 6. y = x + 2 7. y = 3(x + 1)

8. y = -x2 - 3 9. y = 2x - 1 10. y = 1 - 3x2

11. y = 5x2 12. y = (x + 3)2 13. y = 6x2 - 4

14. y = 3x2 - 2 15. y = (x + 4)2 - 4 16. y = -x2 + 4

Graph each relation and its inverse.

17. y = x + 33 18. y = 1

2 x + 5 19. y = 2x + 5

20. y = 12 x2 21. y = (x + 2)2 22. y = (2x - 1)2 - 2

�2

2

�2 2

y

x

�2

�4

2

4

�2 2 4

y

x

y

−1 0 1x

−3 −1 1 2

3

y

3 2 1x

−3 −2 −1 0

0

y = t1x − 2; no

y = t1−x − 3; no

y = t2x5; no

y = t2x + 23

; no

y = x − 2; yes

y = 12x + 1

2; yes

y = t1x − 3; no

y = t1x + 4 − 4; no

y = 13 x − 1; yes

y = t21 − x3

; no

y = t2x + 46 ; no

y = t14 − x; no

2 4 6�4

246

O

y

x2 6�4�6 �2

2

�2�4�6

46

O

y

x4�4�6 �2

2

�2�4�6

xO

y

4

4�4

�4

y

O x �6 �2 246

2

y

x

�2�3 21 3�2�3

123

O

y

x

�4

2

4

�4 2 4

y

x

�2

2

�2 2

y

x

y

23 21 0x

0 1 2 3

22

y

−3 −2 −1x

−2 −1 0 1

0


Name Class Date

For each function, find the inverse and the domain and range of the function and its inverse. Determine whether the inverse is a function.

23. f (x) = 16 x 24. f (x) = -1

5 x + 2 25. f (x) = x2 - 2

26. f (x) = x2 + 4 27. f (x) = 1x - 1 28. f (x) = 13x

29. f (x) = 3 - x 30. f (x) = (x + 1)2 31. f (x) = 11x

32. The equation f (x) = 198,900x + 635,600 can be used to model the number of utility trucks under 6000 pounds that are sold each year in the U.S. with x = 0 representing the year 1992. Find the inverse of the function. Use the inverse to estimate in which year the number of utility trucks under 6000 pounds sold in the U.S. will be 6,000,000. Source: www.infoplease.com

33. The formula s = 0.04n + 2500 gives an employee’s monthly salary s, in dollars, after selling n dollars in merchandise at an appliance store.

a. Find the inverse of the function. Is the inverse a function? b. Use the inverse to find the amount of merchandise sold if the employee’s

salary was $2820 last month.

34. The formula for the surface area A of a sphere of radius r is A = 4pr2 for r Ú 0. a. Find the inverse of the formula. Is the inverse a function? b. Use the inverse to find the radius of a sphere with surface area 10,000 m3.

Let f (x) = 2x + 5. Find each value.

35. ( f -1 ∘ f )(-1) 36. ( f ∘ f -1)(3) 37. ( f ∘ f -1)(- 12 )



$8000

n = 25s − 62,500; yes

f −1(x) = 6x; The domain and range of f and f −1 is the set of all real numbers; f −1 is a function.

f −1(x) = 3 − x; the domain and range of f and f −1 is the set of all real numbers; f −1 is a function.

f −1(x) =−1 t 1x; domain f = all real numbers = range f −1; range f = all nonnegative real numbers = domain f −1; f −1 is not a function.

f −1(x) =t1x − 4; Domain of f =all real numbers = range of f −1; Range of f = all real numbers greater than or equal to 4 = domain of f −1; f −1 is not a function.

f −1(x) =x2 + 1; Domain of f = all real numbers greater than or equal to 1 = range of f −1; Range of f = all real numbers greater than or equal to 0 = domain of f −1; f −1 is a function.

f −1(x) = −5x + 10; The domain and range of f and f −1 is the set of all real numbers; f −1 is a function.

f −1(x) = t1x + 2; Domain of f = all real numbers = range of f −1; Range of f = the set of real numbers greater than or equal to −2 = domain of f −1; f −1 is not a function.

f −1(x) = 13 x2; The

domain and range of f and f −1 is the set of all real numbers greater than or equal to 0; f −1 is a function.

f −1(x) = 1x2; the domain and

range of f and f −1 is the set of all positive real numbers; f −1 is a function.

f −1(x) = x − 635,600198,900 in 2019

r = 2 A4p

; yes

28.2 m

−12−1 3


Name Class Date

Practice Form K


Find the inverse of each relation. Graph the given relation and its inverse.

1. 2. 3.

Find the inverse of each function. Is the inverse a function? To start, switch x and y.

4. y = x2 5. y = x2 + 4 6. y = (3x - 4)2

x = y2

Graph each relation and its inverse.

7. y = 3x - 4 8. y = -x2 9. y = (3 - 2x)2

x y

0

1

2

3

−1

1

3

5

x y

−2

0

2

4

7

3

7

19

x y

−3

−2

−1

0

2

2

2

2

y = 2x; yes y = t1x - 4; noy = t1x + 4

3 ; no

x

y4

2

�2�4�2

�4

42Ox

y

O

4

2

�2�2

�4

42

xO

y

4

6

2

4 62

x y

−1135

0123

x

y

2

4

6

2 4 6O

x y

737

19

−2024

x

y

5

10

15

5 10 15O

x y

2222

−3−2−1

0

x

y4

�2�4�2

�4

4O


Name Class Date



Find the inverse of each function. Is the inverse a function?

10. f (x) = (x + 1)2 11. f (x) = 2x3

5 12. f (x) = 13x + 4

13. Multiple Choice What is the inverse of y = 5x - 1?

f -1(x) = 5x + 1 f -1(x) = x + 15 f -1(x) = x

5 + 1 f -1(x) = x5 - 1

For each function, find its inverse and the domain and range of the function and its inverse. Determine whether the inverse is a function.

14. f (x) = 1x + 1 15. f (x) = 10 - 3x 16. f (x) = 4x2 + 25

17. The formula for the area of a circle is A = pr2. a. Find the inverse of the formula. Is the inverse a function? b. Use the inverse to find the radius of a circle that has an area of 82 in.2.

For Exercises 18−20, f (x) = 5x + 11. Find each value. To start, rewrite f (x) as y and switch x and y.

18. 1 f 5 f -12(5) 19. 1 f -1 5 f2(-3) 20. 1 f -1 5 f2(0)

y = 5x + 11

y = t1x − 1; no y = 53 52 x; yes y = (x − 4) 2

3 ; yes

B

5 −3 0

f −1(x) = x 2 − 1; domainf(x): x # −1, range f(x): y # 0; domain f −1: x # 0, range f −1: y # −1; the inverse is a function.

f −1(x) = −13 x + 10

3 ; domain f(x): all real numbers; range f(x): all real numbers; domain f −1: all real numbers; range f −1: all real numbers; the inverse is a function.

f−1(x) = t1x − 252 ; domain

f(x): all real numbers; range f(x): y # 25; domain f −1: x # 25; range f −1: all real numbers; the inverse is not a function.

r = 5Ap ; yes

about 5.1 in.


Name Class Date

Multiple Choice


1. What is the inverse of the relation? y 3 1 −1

x −2 −1 0 2

−2

y 2 0 −1

x −2 −1 1 3

−2

y −2 −1 1

x −2 −1 0 2

3

y �3 �1 1

x �2 �1 0 2

2

y 2 1 −1

x −2 −1 1 3

−2

2. What is the inverse of the function? y = 5(x - 3)

y = x + 35 y = 1

5 x + 3 y = 5(x + 3) y = 15 x - 3

3. What function with domain x Ú 5 is the inverse of y = 1x + 5?

y = x2 + 5 y = x2 - 5 y = (x - 5)2 y = (x + 5)2

4. What is the domain and range of the inverse of the function? y = 1x - 5

domain is the set of all real numbers Ú 0; range is the set of all real numbers Ú 5

domain is the set of all real numbers Ú 5; range is the set of all real numbers Ú 0

domain and range is the set of all real numbers Ú 5

domain and range is the set of all real numbers

Extended Response

5. A high school principal uses the formula y = 150x + 180 to predict a student’s score on a state achievement test using the student’s 11th-grade GPA number x.

a. What is the inverse of the formula? b. Is the inverse a function? c. Using the inverse, what GPA does a student need to get a passing score of

510 on the state exam?

Standardized Test Prep Inverse Relations and Functions

A

G

C

F

[4] a. y = 1150 (x − 180) or y = 1

150 x − 65 b. yes c. 2.2 [3] most work is correct but

there are minor errors [2] student understands the problem and shows some correct work [1] student may understand the problem but doesn’t know how to proceed OR correct answers without work shown [0] no answers given


Name Class Date

Composition, Inverses, and Linear Functions

Solving an equation for one variable in terms of another is an important step in finding inverses. This step is also used in conversion formulas.

Consider the following linear functions. Let F denote the temperature in degrees Fahrenheit, C the temperature in degrees Celsius, and K the temperature in degrees Kelvin. The formula for converting degrees Fahrenheit to degrees Celsius is C = 5

9 (F - 32), and the formula for converting degrees Celsius to degrees Kelvin is K = C + 273.

1. Use composition to determine the formula for converting degrees Fahrenheit to degrees Kelvin.

2. Solve this function for F.

3. This new equation converts degrees to degrees .

4. Derive a formula to convert degrees Celsius to degrees Fahrenheit.

5. Derive a formula to convert degrees Kelvin to degrees Celsius.

6. Compose these two functions to find a formula for converting degrees Kelvin to degrees Fahrenheit.

Solve each of the following problems involving functions.

7. In 1940, the cost of a new house was $10,000. By 1980, this cost had risen to $90,000. Assuming that the increase is linear, find a function expressing the cost c of a new house in terms of the year y. Solve this function for y. What does this new function enable you to do?

8. Between the ages of 5 and 15, a typical child grows at a fixed annual rate. If Mary was 42 in. in height when she was 5 yr old and grew at a rate of 2 in. a year, find a formula that expresses Mary’s height h in inches when her age is a years. Solve this function for a. What does this new function enable you to do?

9. The air temperature, in degrees Fahrenheit, surrounding an airplane on one particular day was modeled by T = - 1

200a + 110, where a is the altitude, in feet, of the airplane. Solve this function for a. What does this new function enable you to do?

10. The formula L = 0.25W + 0.5 models the length of a certain spring, in inches, when a weight of W ounces is attached to it. Solve this function for W. What does this new function enable you to do?

EnrichmentInverse Relations and Functions

K = 5(F − 32)9 + 273

F = 95 (K − 273) + 32

F = 95 (K − 273) + 32

Kelvin Fahrenheit

C = K − 273

a = 200(−T + 110); find the altitude of the airplane given the temperature

F = 95C + 32

c = 10,000 + 2000( y − 1940); y = c − 10,0002000 + 1940; find the year given the cost of a house

h = 2(a − 5) + 42; a = h − 422 + 5; compute Mary’s age given her height

W = L − 0.50.25 ; find the weight attached to the spring given the length of the stretched

spring


Name Class Date

• Inverseoperations“undo”eachother.Additionandsubtractionareinverse operations. So are multiplication and division. The inverse of cubing a number is taking its cube root.

• Iftwofunctionsareinverses,theyconsistofinverseoperationsperformedin the opposite order.

Problem

What is the inverse of the relation described by f (x) = x + 1?

f (x) = x + 1

y = x + 1 Rewrite the equation using y, if necessary.

x = y + 1 Interchange x and y.

x - 1 = y Solve for y.

y = x - 1 The resulting function is the inverse of the original function.

So, f -1 (x) = x - 1.

Exercises

Find the inverse of each function.

1. y = 4x - 5 2. y = 3x3 + 2 3. y = (x + 1)3

4. y = 0.5x + 2 5. f (x) = x + 3 6. f (x) = 2(x - 2)

7. f (x) = x5 8. f (x) = 4x + 2 9. y = x

10. y = x - 3 11. y = x - 12 12. y = x3 - 8

13. f (x) = 1x + 2 14. f (x) = 23 x - 1 15. f (x) = x + 3

5

16. f (x) = 2(x - 5)2 17. y = 1x + 4 18. y = 8x + 1

ReteachingInverse Relations and Functions

f−1 = x + 54

f−1 = 2x − 4

f −1(x) = 5x

f−1 = x + 3

f−1 (x) = x2 − 2 for x # −2

f−1 = 32x − 23

f −1(x) = x − 3

f −1(x) = x − 24

f−1 = 2x + 1

f −1(x) = 32(x + 1)

f−1 = 32x − 1

f −1(x) = x + 42

f−1 = x

f−1 = 32x + 8

f −1(x) = 5x − 3

f −1(x) = 5 t 2x2 f−1 = (x − 4)2 for x # 0 f−1 = x − 1

8


Name Class Date

Examine the graphs of f (x) = 1x - 2 and its inverse, f -1(x) = x2 + 2, at the right.

Notice that the range of f and the domain of f -1 are the same: the set of all real numbers x Ú 0.

Similarly, the domain of f and the range of f -1 are the same: the set of all real numbers x Ú 2.

This inverse relationship is true for all relations whenever both f and f -1 are defined.

Problem

What are the domain and range of the inverse of the function f (x) = 13 - x?

f is defined for 3 - x Ú 0 or x … 3.

Therefore, the domain of f and the range of f -1 is the set of all x … 3.

The range of f is the set of all x Ú 0. So, the domain of f -1 is the set of all x Ú 0.

Exercises

Name the domain and range of the inverse of the function.

19. y = 2x - 1 20. y = 2 - 1x 21. y = 1x + 5

22. y = 1-x + 8 23. y = 31x + 2 24. y = (x - 6)2

25. y = x2 - 6 26. y = 1x + 4 27. y = 1

(x + 4)2

Reteaching (continued) Inverse Relations and Functions

f �1

f

6

6

5

5

4

4

3

3

2

2

1

O 1

y

x

f

6

3

5

2

4

1

3

2

–1

1

–2–3

y

xO

The domain and the range is the set of all real numbers.

domain: x # 8; range: y " 0

domain: x # −6; range: all real numbers

domain: x ≠ 2; range: y ≠ 0

domain: x # 2; range: y # 0

domain: x ≠ 0; range: y ≠ −4

domain: x # 0; range: y # −5

domain: x # 0; range: all real numbers

domain: x + 0; range: y ≠ −4


Name Class Date

Additional Vocabulary Support Graphing Radical Functions

1. Circle the radical functions in the group below.

y = 2(x + 5) - 3 y = -21x - 4 y = x - 4 y = 13 x + 8

2. Circle the square root functions in the group below.

y = 13 x - 2 y = 1x + 4 y = 21x - 3 y = 3(x + 6)

For Exercises 3−8, draw a line from each word or phrase in Column A to its matching item in Column B.

Column A Column B

3. parent function A. y = 1x - k

4. translate k units downward B. y = - 1x

5. stretch vertically by the factor k (k 7 1) C. y = 1x - k

6. translate k units upward D. y = 1x

7. reflection in x-axis E. y = k1x

8. translate k units to the right F. y = 1x + k

Identify the meaning of the following terms in the function y = 21x − 4 + 5.

9. 2: .

10. 4: .

11. 5: .

Stretch vertically by a factor of 2

Translate 4 units to the right

Translate 5 units upward


Name Class Date

Think About a Plan Graphing Radical Functions

Electronics The size of a computer monitor is given as the length of the screen’s diagonal d in inches. The equation d = 5

613A models the length of a diagonal of a monitor screen with area A in square inches.

a. Graph the equation on your calculator.

b. Suppose you want to buy a new monitor that has twice the area of your old monitor. Your old monitor has a diagonal of 15 inches. What will be the diagonal of your new monitor?

1. How can you use a graph to approximate the area of the old monitor?

.

2. Graph the equation on your calculator. Make a sketch of the graph.

3. What is the area of the old monitor?

4. How can you check your answer algebraically?

.

5. Show that your answer checks.

6. How can you find the diagonal of a new monitor with twice the area of the old monitor?

.

7. Use your method to find the diagonal of your new monitor.

8. What will be the diagonal of your new monitor?

0 100 300200 4000

10

20

30

40

Graph the equation and graph y = 15. The x-coordinate of their intersection will be

the area of the old monitor

Substitute 15 for d and solve the equation for A

Substitute 2 times the area of the old monitor for A in the equation

15 = 5623A, 15(6

5) = 23A, 18 = 23A, 182 = 3A, 324 = 3A, A = 108

d = 5623A = 5

623 # 2 # 108 = 562648 ? 21.2 in.

108 in.2

about 21.2 in.

d

A

Name Class Date


Practice Form G

Graphing Radical Functions

Graph each function.

1. y = 1x + 3 2. y = 1x - 1 3. y = 1x + 5

4. y = 1x - 3 5. y = -21x - 2 6. y = 141x - 1 + 5

Solve each square root equation by graphing. Round the answer to the nearest hundredth, if necessary. If there is no solution, explain why.

7. 1x + 6 = 9 8. 14x - 3 = 5 9. 13x - 5 = 11 - x

10. If you know the area A of a circle, you can use the equation r = 2Ap to find

the radius r. a. Graph the equation. b. What is the radius of a circle with an area of 350 ft2?


11. y = - 13 x + 2 12. y = 213 x - 3 13. y = 13 x + 3 - 1

2

2

4

4

6

6

y

xO

8

8

O

2

�2

�2

2 x

y

2

2

4

4 6

y

Ox

6

8

2

�22

4

4 6

y

xO

6

8

O

2

�2

�2

2 x

y

�8

�4

�6

�22 4 6

y

O x

�4 �2 2

4

1

�2

y

xO

6

O

2

�2�4

�2

2 x

y

2

2

4

4

6

6

y

x

8

8O

75 7no solution; 32 is extraneous

about 10.6 ft

A

r

O 100 200

4

8

300

12

Name Class Date


Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph.

14. y = 181x + 162 15. y = - 14x + 20 16. y = 13 125x - 250

17. y = - 164x + 192 18. y = - 13 8x - 56 + 4 19. y = 125x + 75 - 1

20. y = 10.25x + 1 21. y = 5 - 14x + 2 22. y = 13 27x - 54

23. To find the radius r of a sphere of volume V, use the equation r = 53 3V4p.

a. Graph the equation. b. A balloon used for advertising special events has a volume of 225 ft3.

What is the radius of the balloon?

24. An exercise specialist has studied your exercise routine and says the formula t = 1.851c + 10 expresses the amount of time t, in minutes, it takes you to burn c calories (cal) while exercising.

a. Graph the equation. b. According to this formula, how long should it take you to burn

100 cal? 200 cal? 300 cal?

25. You can use the equation t = 141d to find the time t, in seconds,

it takes an object to fall d feet after being dropped. a. Graph the equation. b. How long does it take the object to fall 400 feet?



y = 0.51x + 4; graph of y = 0.51x shifted left 4 units

y = 91x + 2; graph of y = 91x shifted left 2 units

y = −81x + 3; graph of y = −81x shifted left 3 units

y = 5 − 22x + 12; graph

of y = −21x shifted left 12 unit and up 5 units

y = −21x + 5; graph of y = −21x shifted left 5 units

y = −2 31x − 7 + 4; graph of y = −2 31x shifted right 7 units and up 4 units

y = 3 31x − 2; graph of y = 3 31x shifted right 2 units

y = 5 31x − 2; graph of y = 5 31x shifted right 2 units

y = 51x + 3 − 1; graph of y = 51x shifted left 3 units and down 1 unit

3.77 ft

19.4 min, 26.8 min, 32.6 min

5 s

2

100

3

1

54

500300

r

VO

20

100O

30

10

5040

400

t

c

4

200O

6

2

10 t8

800

d

Name Class Date


Practice Form K



1. y = 1x + 3 2. y = 1x - 4 3. y = 1x - 7


4. y = 41x 5. y = -21x + 1 6. y = 51x - 4

Solve each square root equation by graphing. Round the answer to the nearest hundredth, if necessary. If there is no solution, explain why.

7. 1x + 2 = 7 8. 14x + 1 = 5 9. 313 - x = 10

10. A periscope on a submarine is at a height h, in feet, above the surface of the water. The greatest distance d, in miles, that can be seen from the periscope on a clear day is given

by d = 53h2 .

a. If a ship is 3 miles from the submarine, at what height above the water would the submarine have to raise its periscope in order to see the ship?

b. If a ship is 1.5 miles from the submarine, to what height would it have to be raised?

xO

y

4

6

2

4 62

xO

y

4

6

2

4 62

xO

y

�2

�4

�6

4 62

xO

y

2

4

6

4 62

xO

y

�6

�4

�24 62

xO

y

2

�2

�4

4

4 62

xO

2468

20 30 4010

y

xO

y

2

4

6

4 62

y

3

6

9

47 6 −8.11

6 ft1.5 ft

Name Class Date




Graph each function. To start, graph the parent function, y = 13 x.

11. y = 13 x - 4 12. y = 3 - 13 x + 1 13. y = 1213 x - 1 + 3

14. A center-pivot irrigation system can water from 1 to 130 acres of crop land. The length l in feet of rotating pipe needed to irrigate A acres is given by the function l = 117.751A.

a. Graph the equation on your calculator. Make a sketch of the graph. b. What length of pipe is needed to irrigate 40, 80, and 130 acres?

Graph each function. Find the domain and range.

15. y = 213 x - 4 16. y = - 13 8x + 5 17. y = -31x - 4 - 3

18. Open Ended Write a cube root function in which the vertical translation of y = 13 x is twice the horizontal translation.

function in the form y = a 31x − h + k where k = 2h.Answers may vary. Any cube root

x

y

�2�2

�4

�6

2

x

y6

�2

4

2 x

y6

�2

4

2

744.7 ft, 1053.2 ft, 1342.6 ft

xO

y4

2

2 4 64

�8

812

2�2 xO

y

�4�2

�8�6

24

24 6 8

xO

y

domain: all real numbers; range: all real numbers

domain: all real numbers; range: all real numbers

domain: x # 4; range: y " −3

xO

y300

�2�4

200

100

2 4


Name Class Date

Multiple Choice


1. What is the graph of y = 1x + 4?

2. What is the graph of y = 1x - 3 - 2?

3. What is the graph of y = 1 - 13 x + 3?

4. What is the description of y = 19x - 3 to make it easy to graph using transformations of its parent function?

the graph of y = 31x, shifted right 3 units

the graph of y = 31x, shifted right 13 unit

the graph of y = 1x, shifted right 3 units and up 9 units

the graph of y = 1x, shifted right 13 unit and up 9 units

Short Response

5. What is the graph of y = 21x - 1 + 3?

Standardized Test Prep Graphing Radical Functions

4

2�2�4

68

4

y

xO

42

1

�2�3

8

y

xO

�2�2�4�6

�4

4y

xO

4

642O

6

2

8

8

y

x

�2 2�4

2

�4

4

4

y

x

2�2�4

2

�2�4

4

4

y

x

4

642O

6

2

8

8

y

x

642

2

�4

46

8

y

xO

�2�4�6�8

2

�4

4y

xO

4

2�2�4

68

4

y

xO

4

642O

6

2

8

8

y

x

�2�2 2�4�6

2

�4

4y

x

C

F

D

G

[2] The graph is correct.[1] One of the transformations (horizontal, vertical, or stretch) is incorrect.[0] no answer given

642O

4

2

6 y

x


Name Class Date

EnrichmentGraphing Radical Functions

Transformations of Other FunctionsYou can obtain the graph of any function of the form y = a # f (x - h) + k by using the shifting rules similar to those used to obtain the graph of y = a1x - h + k. Note that the second function is a special case of the first when f (x) = 1x. To obtain the graph of y = a # f (x - h) + k, given the graph of y = f (x), use the following general rules:

• Ifa 6 0, reflect the graph of y = f (x) across the x-axis.

• If 0a 0 7 1, the graph of y = f (x) is stretched by a factor of a.

• If0 6 0a 0 6 1, the graph of y = f (x) is compressed by a factor of a.

• Thegraphofy = f (x) is shifted right h units if h 7 0 and left h units if h 6 0.

• Thegraphofy = f (x) is shifted up k units if k 7 0 and down k units if k 6 0.

1. Use the general rules to describe how the graph of y = -3(x - 5)2 + 7 can be obtained from the graph of f (x) = x2.

2. Write the equation for the graph that looks like y = 13 x but that is shifted right four units, reflected across the x-axis, and shifted down six units.

3. Use the graph of y = f (x) given below to sketch the graph of y = f (x + 2) - 1.

4. The graph of y = f (x) and y = g(x) is given below. The graph of g is a transformation of the graph of f. Write the equation for the graph of g in terms of f.

O

4

�4

�4

4

x

y

y � f (x)

2

�2

�2

2

x

y

f

g

O

O

42

6

�4�6 �2

�4�6

42 6

x

y

reflect across the x-axis, stretch by a factor of 3, shift right 5 units and shift up 7 units

y = − 31x − 4 − 6

g(x) = f(x + 1) − 3

O

4

�4

�4

4

x

y


Name Class Date

ReteachingGraphing Radical Functions

The graph of y = a1x - h + k is a translation h units horizontally and k units vertically of y = a1x. The value of a determines a vertical stretch or compression of y = 1x.

Problem

What is the graph of y = 21x - 5 + 3?

y = 21x - 5 + 3

a = 2 h = 5 k = 3

Translate the graph of y = 21x right five units and up three units. The graph of y = 21x looks like the graph of y = 1x with a vertical stretch by a factor of 2.

Exercises


1. y = 1x - 4 + 1 2. y = 1x - 4

3. y = 1x + 1 4. y = - 1x + 2 - 3

5. y = 21x - 1 6. y = -21x + 3 + 4

7. y = - 1x + 1 8. y = 1x + 3 - 4

9. y = 31x + 2 10. y = - 1x - 2

O

2

4

6

62 4x

y

O

2

4

62 4x

y

O

2

�2�2

2x

y

O

2

�2�2

2x

y

O�2�2

�4

2xy

O�2

�2�4xy

O

2

4

6

�2�4x

y

O

2

�2

�4

4x

y

O

2

4

6

2 4x

y

O�2

�2�4

�4

xy


Name Class Date



Graphs can be used to find solutions of equations containing radical expressions.

Problem

What is the minimum braking distance of a bicycle with a speed of 22 mph?

You can find the minimum braking distance d, in feet, of a bicycle travelling s miles per hour using the equation s = 5.51d + 0.002.

We want to find the value of d when s = 22. In other words, solve the equation 5.51d + 0.002 = 22. Graph Y1 = 5.52(X + 0.002) and Y2 = 22. Try different values until you find an appropriate window. Then use the intersect feature to find the coordinates of the point of intersection.

The minimum braking distance will be about 16 ft.

Exercises

Solve the equation by graphing. Round the answer to the nearest hundredth, if necessary. If there is no solution, explain why.

11. 13x + 1 = 5 12. 14x + 1 = 9

13. 12 - 5x = 4 14. 13x + 5 = 7

15. 17x + 2 = 11 16. 12x - 1 = 11 - 2x

17. 1x - 2 = 12 - 3x 18. 71x - 3 = 212x + 1

19. 12x - 5 = 14 - x 20. 12x + 7 = 315x + 2

Y � 22Intersectionx � 15.998

8 20

14.67

0.5

3.68

−0.26

17

3

no solution; x = 1 is extraneous

−2.8

Documents

Chapter 6 Performance Tasks - BergSSMathbergssmath.weebly.com/uploads/4/2/4/6/4246450/6-4...[2] Correct answer with no work shown OR student only able to complete part of Task. [1]