Polynomials Day 2 Inverse/compositions

Even and odd functionsSynthetic Division

Characteristics

InversesMA2A2. Students will explore inverses of functions.

Discuss the characteristics of functions and their inverses, including one-to-oneness, domain, and range.Determine inverses of linear, quadratic, and power functions and functions of the form , including the use of restricted

domains.Explore the graphs of functions and their inverses.

Use composition to verify that functions are inverses of each other.

A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain.

A function is not one-to-one if two different elements in the domain correspond to the same element in the range.

Theorem Horizontal Line Test

If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.

Use the graph to determine whether the function f x x x( ) 2 5 12

is one-to-one.

Not one-to-one.

Use the graph to determine whether the function is one-to-one.

One-to-one.

The inverse of a one-one function is obtained by switching the role of x and y

Let and

Find

3)( xxf 31

)( xxg

))(())(( xfgandxgf

g is the inverse of f.

311 )( xxf

Domain of f Range of f

Range of f 1 Domain of f 1

f 1

f

Domain of Range of

Range of Domain of

f f

f f

1

1

Theorem

The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

f 1

2 0 2 4 6

2

2

4

6 f

f 1

y = x

(2, 0)

(0, 2)

Finding the inverse of a 1-1 function

Step1: Write the equation in the form

Step2: Interchange x and y.

Step 3: Solve for y.

Step 4: Write for y.

)(xfy

)(1 xf

Find the inverse of

Step1:

Step2: Interchange x and y

Step 3: Solve for y

3

5

xy

3

5

yx

3

5)(

xxf

x

xxf

35)(1

Even and Odd Functions

MA2A3. Students will analyze graphs of polynomial functions of higher degree. b. Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and

multiplicity of real zeros.c. Determine whether a polynomial function has symmetry and whether it is even, odd, or neither.

d. Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative and absolute extrema, intervals of increase and decrease, and end behavior.

Even functions

A function f is an even function if

for all values of x in the domain of f.

Example: is even because

)()( xfxf

13)( 2 xxf

)(131)(3)( 22 xfxxxf

Odd functions

A function f is an odd function if

for all values of x in the domain of f.

Example: is odd because

)()( xfxf

xxxf 35)(

)()5(5)(5)( 333 xfxxxxxxxf

Determine if the given functions are even or odd

23

3

24

)()4

1||)()3

)()2

1)()1

xxxk

xxh

xxg

xxxf

Graphs of Even and Odd functions

The graph of an even function is symmetric with respect to the y-axis.

The graph of an odd function is symmetric with respect to the origin.

x

0 4 0

1 3 -1

-1 3 1

2 0 4

-2 0 -4

24)( xxf xxxg 2)( 3

3210-1-2-3

6

5

4

3

2

1

0

-1

-2

-3

-4

-5

-6

x

y

x

y

52.50-2.5-5

5

2.5

0

-2.5

-5

x

y

x

y

Determine if the function is even or odd?

Determine if the function is even or odd?

52.50-2.5-5

5

3.75

2.5

1.25

0

x

y

x

y

52.50-2.5-5

100

50

0

-50

-100

x

y

x

y

Determine if the function is even or odd?

Synthetic DivisionMA2A3. Students will analyze graphs of polynomial functions of higher degree.

b. Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and multiplicity of real zeros.

c. Determine whether a polynomial function has symmetry and whether it is even, odd, or neither. d. Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros,

relative and absolute extrema, intervals of increase and decrease, and end behavior.

3 2

2

3x x 4x 1

x 1

2 3 2x 1 3x x 4x 1

3

2

3x

x3x

3x

33x 3x2x x 1

2 3 2x 1 3x x 4x 1

3x

33x 3x2x x 1 2

21

x

x

1

2x 1x

2

x3x 1

x 1

4 2

2

x 2x x 3

x x 1

2 4 230xx x 1 x 2x x 3

4

22x

xx

4x 3x3 2x x x

2x x

2x

2x

2 4 3 2x x 1 x 0x 2x x 3 4x 3x

3 2x x x

2x

2x

3

2x

x

x

x

3x 2x x

3 2x 2 x 0x x 2 3x

22x x 2

2x

22x

3x x 2

x 2

3 20x 2 x 2xx

23x

xx

3x22x x 2

2x

22x

22x

x2x

2x

22x 4x5x 25x

x5

5

5x 10

12

2 12x 2x 5

x 2

3 2x 4x 2x 5

x 3

3 2x 3 x 4x 2x 5

23x

xx

3 2x 3x2x 2x 5

2x

2

xx

x

x

2x 3x

x 5

1x

x

1

x 3 8

2 8x x 1

x 3

3 22x x 2x 3

x 1

3 2x 1 2x x 2x 3 3

22x

x2x

3 22x 2x23x 2x 3

22x

23x

x3x

3x

23x 3x

x 3

1x

x

1

x 1 4

2 42x 3x 1

x 1

Synthetic Division Summary

1. Set denominator = 0 and solve (box number)2. Bring down first number3. Multiply by box number and add until finished4. Remainder goes over divisor

Notes of Caution

1. ALL terms must be represented (even if coefficient is 0)2. If box number is a fraction, must divide final answer by the denominator

To evaluate a function at a particular value, you may EITHER:A) Substitute the value and simplify ORB) Complete synthetic division…the remainder is your answer

3 2x 4x 2x 5

x 3

x

x

3

3

0

1 -4 2 -5 3

1

3

-1

-3

-1

-3

-8

2x x 1x

8

3

3 22x x 2x 3

x 1

x

x

1

1

0

2 -1 2 -3 1

2

2

1

1

3

3

0

22x x 3

3 24x 3x 8x 4

x 3

x

x

3

3

0

4 -3 -8 4 3

4

12

9

27

19

57

61

24x 9x 161

9x 3

3 22x 5x 28x 14

x 5

x

x

5

5

0

2 -5 -28 14 5

2

10

5

25

-3

-15

-1

22x 5xx

13

5

3 216x 32x 81x 162

x 2

x

x

2

2

0

16 -32 -81 162 2

16

32

0

0

-81

-162

0

216x 81

3 2x 2x x 1

x 3

x

x

3

3

0

1 -2 -1 1 3

1

3

1

3

2

6

7

2x x 2x

7

5

3x 5x 2

x 3

x

x

3

3

0

1 0 -5 2 3

1

3

3

9

4

12

14

2x 3x 4x

4

3

1

4 2x 17x 16

x 4

x

x

4

4

0

1 0 -17 0 16 4

1

4

4

16

-1

-4

-4

3 2x 4x x 4

23x 5

x

0 x 2

3

x

4 23x 17x 1x 60

4

0x

x

-16

0

3 26x 4x 3x 2

3x 2

3x 2 0

2

3x

6 -4 3 -2 2/3

6

4

0

0

3

2

0

22x 1

4 3 22x 5x 4x 5x 2

2x 1

2x 1 0

1x

2

2 5 4 5 2

2

-1

4

-2

2

-1

4

3 2x 2x x 2

-2

0

3

-1/2

2

3 24x x 4x 1

4x 1

4x 1 0

1

4x

4 -1 -4 1 1/4

4

1

0

0

-4

-1

0

2x 1

4

3 21Find f if f x 4x x 4x 1

4

1f4

0

34x 13x 6

2x 1

2x 1 0

1x

2

4 0 -13 -6

4

-2

-2

1

-12

6

0

22x x 6

-1/2

2

31Find f if f x 4x 13x 6

2

f 01

2

3

x 3

x 5x 2

x 3 0

x 3

1 0 -5 2 3

1

3

3

9

4

12

14

2 14x 3x 4

x 3

3 2x 0x 5x 2

x 3

3Find f 3 if f x x 5x 2

f 3 14

4 2x 17x

x 4

16

x 4 0

x 4

1 0 -17 0 16 4

1

4

4

16

-1

-4

-4

3 2x 4x x 4

4 3 2x 0x 17x 0x 16

x 4

-16

0

4 2Find f 4 if f x x 17x 16

f 4 0

Direct Substitution

A fancy term for plug in and find f(x).

Homework- Math 3 Book

Page 69◦1-19 all, skip 8 – 10 and don’t do

part e on 17-19.