+

Inverse FunctionsHow do we verify and find inverses of functions?

M2 Unit 5a: Day 3

+Inverse functions:

Two functions are inverses of one another if

The function g is denoted by , read as “f inverse.”

To verify two functions are inverses, use compositions

f 1

f (g(x)) x

g( f (x)) x

and

+Watch me as I work one.Ex: Verify that functions are inverses

Verify that and are inverses.

1 3( )

4 4g x x= -

f (x)4x 3

1 3( ( )) 4 3

4 4f g x x

æ ö÷ç= - +÷ç ÷çè ø 3 3x= - + x=1 3

( ( )) (4 3)4 4

g f x x= + -3 3

4 4x= + - x=

+You try:

Verify that f(x) and g(x) are inverse functions.

1. f (x) 5x 2 -=

-( 2)

( )5

xg x

æ ö- ÷ç=- + =÷ç ÷çè ø-( 2)

( ( )) 5 25

xf g x x

- + -= =

-5 2 2

( ( ))5

xg f x x

+You try:

Verify that f(x) and g(x) are inverse functions.

2. f (x)4x

=4

( )g xx

= = =4

( ( )) ( ( ))4

f g x g f x x

x

+You try:

Verify that f(x) and g(x) are inverse functions.

3. ( ) 3 6f x x=- + =- +1

( ) 23

g x x

æ ö÷ç= - + =÷ç ÷çè ø

æ ö÷ç- - + + =÷ç ÷çè ø

- + =

1( ( )) 2

3

13 2 6

3

6 6

f g x f x

x

x x

( )= - + =

- - + + =

- + =

( ( )) 3 6

1( 3 6) 2

32 2

g f x f x

x

x x

+4. Which pair of functions are inverses? (Use composition to determine the answer.)

a.

b.

c.

d.

f(x)=5x -7 and g(x)=5x+7

x+7f(x)=5x -7 and g(x)=

5

5f(x)=5x and g(x)=

x

f(x)=5x and g(x)=x - 5

+5. Which pair of functions are inverses? (Use composition to determine the answer.)

a.

b

c.

d.

1

2f(x)=2x - 4 and g(x)= x+4

x

7f(x)=7x and g(x)=

f(x)=x+4 and g(x)=x - 4

f(x)=2- x and g(x)=2+x

+Inverse Relations

An inverse relation maps the output values back to their original input values.

Basically, switch the domain and range of the relation to find the inverse.

Ex: Find the inverse.

x -2 -1 0 1 2

y 4 2 0 -2 -4

x 4 2 0 -2 -4

y -2 -1 0 1 2

Switch the x and y values

+Ex: 6 Now you try.

Find the inverse of the relation.

a)

b) x 1 3 6 9 12

y -1 -3 -6 -9 -12

x -1 -3 -6 -9 -12

y 1 3 6 9 12

x 3 4 5 6 7

y -4 -2 0 2 4

x -4 -2 0 2 4

Y 3 4 5 6 7

+Example 7:Watch me as I work one:

Find g(x), the inverse of the function.

( ) 4 3f x x= +

= -1 3

( )4 4

g x x

+Example 8:Find the inverse of a function

Consider the function . Find the inverse.

f (x)2xSwitch x and y and solve for y!

2x

y= 2y x =g

2y

x=

+Example 9:

Find g(x), the inverse of the function.

( ) 5 2f x x=- +

( 2)( )

5

xg x

-=

-

+Example 10:

Find the inverse of the function:

f (x) 2x 5

Is there any value that we can’t use for x?

Switch x and y and solve for y!

2 5x y= - 2 2 5x y= - 2 5 2x y+ =

2 5

2

xy

+= , 0x ³

+11. Which of the following is the inverse of ?

a.

b

c.

d.

f (x)12x 10

f 1(x)(x 10)2

f 1(x)(x 10)2

f 1(x)2(x 10)

f 1(x)2(x 10)

+In summary…

How do you read the symbol ?“f inverse”

How do you verify that 2 functions are inverses of one another?Use compositions to show that f(g(x)) = x and g(f(x)) = x.

f 1

+Homework:

4.3 Practice Worksheet

(#1-8, 12-17)