Checkpoint 1 Perform Function Operations · Algebra 2 Honors – Unit 8 – Inverses and Exponential Growth Checkpoints Name Checkpoint 1 – Perform Function Operations Given the

Algebra 2 Honors – Unit 8 – Inverses and

Exponential Growth

Checkpoints

Name

Checkpoint 1 – Perform Function Operations

Given the following functions, find k(x). Use a calculator to find the domain.

1. ( ) ( )k x f x j x 2. ( ) ( )k x f x g x 3. ( ) ( )k x f x h x 4. ( )

( )

g xk x

j x

Domain: Domain: Domain: Domain:

5. f h x 6. h g x 7. f g x 8. /j g x


9. 𝑓(5) 10. 𝑗(−5) 11. 𝑔(−4)

12. ℎ(2) + 1 13. 2𝑔(3) 14. −𝑗(−9)

2 6 9 3 3

xf x x x g x x h x j x x

x

Use the graph of 𝒚 = 𝒇(𝒙) to graph each function of g.

15. 1g x f x

16. 2g x f x 17. g x f x 18. 3g x f x

19. The function 𝑓(𝑥) = 2.9√𝑥 + 20.1 models the

median height, 𝑓(𝑥), in inches, of boys who are

x months of age.

a) According to this model, find the

median height of a boy 48 months old.

b) At what approximate age will a boy be

36 inches tall?

c) Growth Hormone Deficient (GH) can

slow growth in male children. A normal boy 30 months old is about 35.98 inches

tall. A boy with GH will take 40 months to get to the same height. Explain, in

function notation, the translation between the two boys.

Checkpoint 2 – Composition Functions

Find the following composite functions given the following.

1. g f x 2. f h x 3. j g x 4. h j x


5. f j 6. h g 7. f h g x 8. j f g x


9. ( ( 8))h g 10. ( )(6)g h 11. ( )( 1)g j 12. ( (1))j g f

1

2 216 4 4 f x x g x x h x x j x x

13. Let 𝑓(𝑥) = 16𝑥2 3⁄ and 𝑔(𝑥) = 4

𝑥. Find 𝑔(𝑓(𝑥)).

A. 𝑔(𝑓(𝑥)) =

√𝑥3

4𝑥 C. 𝑔(𝑓(𝑥)) =

16 ∙ √16𝑥3

𝑥

B. 𝑔(𝑓(𝑥)) =64 ∙ √𝑥23

𝑥 D. 𝑔(𝑓(𝑥)) =

√16𝑥3

4𝑥

14. Given a function 𝑓(𝑥) = −𝑥0.5 + 2, 𝑔(𝑥) = 𝑥 + 1 , and ℎ(𝑥) = 𝑓(𝑔(𝑥)). Find when ℎ(𝑥) is

positive.

A. [0,3)

B. [−1, ∞)

C. [−1,3)

D. (−∞, 3)

15. The number N of bacteria in refrigerated food is given by the function

𝑁(𝑇) = 20𝑇2 − 80𝑇 + 500, 2 ≤ 𝑇 ≤ 14 where T is the temperature of the food in degrees

Celsius. When the food is removed from refrigeration, the temperature of the food is

given by 𝑇(𝑡) = 4𝑡 + 2, 0 ≤ 𝑡 ≤ 3 where t is in hours.

a) Find the composition 𝑁(𝑇(𝑡)) and interpret its meaning in context.

b) Find the time when the bacteria count reaches 2000.

Checkpoint 3 – Inverse Functions

Find the inverse of the following equations.

1. 2 5y x 2. 10 28y x 3. 2

23

y x 4. 6 11y x

Verify the g and g are inverse functions.

5. 4 4f x x , g x x 6. 1

1 5 55

f x x , g x x

Find the inverse of the power functions. (Hint: make sure you rationalize denominators

and use plus/minus when appropriate).

7. 2 1f x x 8. 24y x 9. 32

5f x x 10. 51

64f x x

11. 2

3 1f x x 12. 430

2f x x ,x 13. 53

54

f x x 14. 32 6

9

xf x

Determine whether the inverse of 𝒇 is a function.

15. 5f x x 16. 26 0f x x ,x 17. 3 2f x x 18. 4f x | x |

19. Determine whether 𝑓(𝑥) = 𝑥 − 3 and 𝑔(𝑥) = −𝑥 + 3 are inverse functions.

Explain.

A. 𝑓(𝑥) and 𝑔(𝑥) are inverse functions because 𝑓(𝑥) + 𝑔(𝑥) = 0

B. 𝑓(𝑥) and 𝑔(𝑥) are inverse functions because 𝑓(𝑔(𝑥)) = −𝑥

C. 𝑓(𝑥) and 𝑔(𝑥) are not inverse functions because 𝑓(𝑥)

𝑔(𝑥)= −1

D. 𝑓(𝑥) and 𝑔(𝑥) are not inverse functions because 𝑓(𝑔(𝑥)) = −𝑥

20. Find the inverse of 𝑓(𝑥) =1

6𝑥3 + 8

A. 𝑓−1(𝑥) = 6𝑥3 − 8 C. 𝑓−1(𝑥) = √6𝑥

3− 2

B. 𝑓−1(𝑥) = √6𝑥 + 83

D. 𝑓−1(𝑥) = √6𝑥 − 483

21. At the start of a dog sled race in Anchorage, Alaska, the temperature was 5 C . By

the end of the race, the temperature was 10 C . The formula for converting

temperatures to Fahrenheit (F) is 5

329

C F .

A. Find the inverse of the given model. What does the inverse describe?

B. Find the difference between the temperatures in the end and beginning of the

race in Fahrenheit.

C. Use a graphing calculator to graph the original functions and its inverse. Find

when the temperature is the same on both temperatures scales.

Checkpoint 4 – Graphing Inverse Functions

Graph the function and the inverse of the function on the following graphs. Write the

inverse function, and its domain and range below. Finally, state if the inverse is a

function.

1. 1

22

f x x 2. 2 6y x

Inverse: _____________ Function? Inverse: ________________ Function?

Domain: _____________ ___________ Domain: _______________ ___________

Range: ________________ Range: ________________

3. 1

3 2f x x 4. 3 5y x


Domain: _____________ ___________ Domain: _______________ ___________

Range: ________________ Range: ________________

5. 2 10 24f x x x 6. 21 1y x x x


Domain: _____________ ___________ Domain: _______________ ___________

Range: ________________ Range: ________________

7. Consider the function g x x .

A. Graph g x x , and its inverse. What do you notice about the two functions?

Relationship?

B. Write another function that it is their own inverse.

Checkpoint 5 – Graphing Exponential Growth Functions

Graph the following functions (include any asymptotes).

1. 3xy 2. 3 2xy 3. 3 2xy

Domain: ____________ Domain: ____________ Domain: ____________

Range: ____________ Range: ____________ Range: ____________

Asymptote Equation: Asymptote Equation: Asymptote Equation:

___________ ___________ ___________

4. 4 2xy 5. 12 3xy 6. 13 4 2xy




___________ ___________ ___________

7. 15 3 4xy 8. 32 5 1xy 9. 5

32

x

y




___________ ___________ ___________

10. The graph of which function is shown?

A. 1

2 1 5x

f x .

B. 2 1 5 1x

f x .

C. 1

3 1 5x

f x .

D. 3 1 5 1x

f x .

11. Which equation is represented by the graph below?

A. 𝑦 = −2 ∙ 4𝑥−2 − 1

B. 𝑦 = −2 ∙ 4𝑥−3 − 1

C. 𝑦 = 2 ∙ 4𝑥−2 − 1

D. 𝑦 = 2 ∙ 4𝑥−3 − 1

Checkpoint 6 – Graphing Exponential Decay Functions

Tell whether the function is exponential growth or decay.

1. 3

34

x

y

2. 5

42

x

f x

3. 4 xf x 4. 0 25x

y .


5. 1

2

x

y

6. 0 25x

y . 7. 1

25

x

y




___________ ___________ ___________

8. 0 5x

y .

9.

31

3

x

y

10.

11

2 33

x

y




___________ ___________ ___________

11.

21

22

x

y

12.

11

34

x

y

13.

51

6 22

x

y




___________ ___________ ___________

14. The graph of which function is shown?

A. 1

2 0 5x

f x .

B. 2 1 5 1x

f x .

C. 1

0 5x

f x .

D. 0 5 1x

f x .

15. Which equation is represented by the graph below?

A. 𝑦 = 3 ∙ (

1

3)

𝑥−1

+ 2

B. 𝑦 = 3 ∙ (1

3)

𝑥

+ 2

C. 𝑦 = 3 ∙ 3𝑥 + 2

D. 𝑦 = −3 ∙ 3𝑥 + 2

Checkpoint 7 – Application of Exponential Growth and Decay Functions

1. In 1992, 1219 monk parakeets were observed in the United States. For the next 11

years, about 12% more parakeets were observed each year.

a) Estimate the number of parakeets in 2000.

b) Estimate the year, in which there were about 6000 parakeets.

2. You purchase an antique table for $450. The value of the table increases by 6% per

year.

a) Estimate the value of the table 5 years later.

b) Estimate how long it will take for the value to double.

3. In 1990, the population of Austin, Texas, was 494,490. During the next 10 years, the

population increased by about 3% each year.

a) What was the population in 2000?

b) Estimate the year when the population was about 590,000

4. A certain medication is eliminated from the bloodstream at a rate of about 12% per

hour. The medication reaches a peak level in the bloodstream of 40 milligrams. Predict

the amount, to the nearest tenth of a milligram, of medication remaining:

a) 3 hours after the peak level b) 5 hours after the peak level

5. Anthrax becomes active when they find the optimal conditions including enough iron

to continue to grow. If a single Anthrax bacterium enters your blood stream; how long

(in hours) will it take for it to reach as many bacteria as cells in a kidney (~509338 cells)?

a) If it doubles every two hours? b) If it doubles every half hour?

6. You buy a mountain bike for $500. The value of the bike decreases by 20% each

year.

a) Estimate the value after 3 years.

b) Estimate when the value of the bike will be $100.

#7-11 Find the final amount for each investment.

7. You deposit $1300 earning 5% annual interest compounded annually for 10 years.

8. You deposit $300 earning 4.5% annual interest compounded quarterly for 3 years.

9. You deposit $2000 earning 2.75% annual interest compounded monthly for 6 years.

10. You deposit $5000 earning 3.5% annual interest compounded daily for 3 years.

11. When will your investment double if you deposit $750 into an account that earns

6.5% interest compounded quarterly?

12. A sample of two bacteria strains are being studied at a lab. After ℎ hours,

the population of Bacteria M is modeled by 𝑀(ℎ) = 20(1.8)ℎ, and Bacteria

N is modeled by 𝑁(ℎ) = 30(1.65)ℎ. When is the population of Bacteria M

greater than the population of Bacteria N?

A. The population of Bacteria M is always greater than the population of

Bacteria N.

B. The population of Bacteria M is never greater than the population of

Bacteria N.

C. The population of Bacteria M is greater until a point between 4 and 5

hours, after which Bacteria N has the greater population.

D. The population of Bacteria N is greater until a point between 4 and 5

hours, after which Bacteria M has the greater population.

Checkpoint 8 – Using Functions involving e

Simplify the expression.

1. 4 3e e 2. 3

32 xe 3. 69e 4. 35x xe e 5. 4

2

6

18

x

x

e

e

Evaluate the following expressions (round to the nearest thousandth).

6. 10 25.e 7. 2 54 /e

Tell whether the function is exponential growth or decay.

8. 3 xy e 9. 41

3

xf x e

10. 0 54 . xf x e


11. 3xy e 12. 42 xy e 13. 24 xy e

Growth / Decay (Circle) Growth / Decay (Circle) Growth / Decay (Circle)




___________ ___________ ___________

14. 0 5 2. xy e 15. 3 2xy e 16. 2 13

xy e

Growth / Decay (Circle) Growth / Decay (Circle) Growth / Decay (Circle)




___________ ___________ ___________

17. Scientist used traps to study the Formosan subterranean termite population in New

Orleans. The mean number y of termites collected annually can be modeled by 𝑦 =738𝑒0.345𝑡 where t is the number of years since 1989. What was the mean number of

termites collected by 2012?

18. One formula for modeling the spread of rumors is 0 41 . t

PH

P S e

where H is the

number of people that heard, P is the total population, S is the number of people who

start he rumor, and t is time in hours. If there are 1600 students in a school, and 2 people

start the rumor; estimate how long it will take for 730 students to hear.

19. If you deposit $800 in an account that pays 2.65% annual interest compounded

continuously. What will the balance be after 12 years?

20. You deposit $3500 in an account that earns 2.5% compounded continuously.

About how long will it take for to double your deposit?

23. Scientists experimenting with the effects of a new antibiotic on a particular

bacteria population found that the a population of bacteria can be modeled with the

function 𝑓(𝑡) = 2000(1 − 0.25)𝑡, where 𝑡 is the time in days the antibiotic is taken.

Scientists have also discovered that this antibiotic can only be taken for a maximum 5

days before it is considered harmful to the patient. In order to consider a person

“cured” of the bacterial infection, an initial population of 2000 bacteria must be

reduced to less than 200. Is it possible to cure a person with the new antibiotic? If so,

how many days will it take to cure the disease?

Documents

Checkpoint 1 Perform Function Operations · Algebra 2 Honors – Unit 8 – Inverses and Exponential Growth Checkpoints Name Checkpoint 1 – Perform Function Operations Given the