Notes 3-8

Section 3-8Inverse Functions

Warm-upIndicate how you would “undo” each operation or

composite of operations.

1. Turn east and walk 50 meters, then turn north and walk 30 meters.

3. Add -70 to a number, then multiply the result by 14.

2. Multiply a number by .45

4. Square a positive number, then cube it.

Inverse of a function:

Inverse of a function:

A function that will “undo” what another function had previously done

Inverse of a function:

A function that will “undo” what another function had previously done

When the independent variable is switched with the dependent variable

Inverse of a function:

A function that will “undo” what another function had previously done

When the independent variable is switched with the dependent variable

**Notation: The inverse of f is f-1

Example 1Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.

a. Find the inverse S-1.

b. Describe S and its inverse in words.

Example 1Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.

a. Find the inverse S-1.

S-1 = {(1,1), (4, 2), (9, 3), (16, 4)}

b. Describe S and its inverse in words.

Example 1Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.

a. Find the inverse S-1.

S-1 = {(1,1), (4, 2), (9, 3), (16, 4)}

b. Describe S and its inverse in words.

S is a squaring function, where the independent variable is squared to obtain the dependent variable.

Example 1Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.

a. Find the inverse S-1.

S-1 = {(1,1), (4, 2), (9, 3), (16, 4)}

b. Describe S and its inverse in words.

S is a squaring function, where the independent variable is squared to obtain the dependent variable.Its inverse is a positive square root function, where you would square root the independent variable to get the dependent variable.

Just a little note:

Just a little note:

As the independent variable switches with the dependent variable, the domain switches with the range.

Theorem(Horizontal-Line Test)

Theorem(Horizontal-Line Test)

If you can draw a horizontal line on the graph of f and it touches the graph more than once, then the

INVERSE of f is not a function.

Theorem(Horizontal-Line Test)

If you can draw a horizontal line on the graph of f and it touches the graph more than once, then the

INVERSE of f is not a function.

The horizontal-line test tells us nothing about the original function...remember that!

Example 2Give an equation for the inverse and tell whether it is

a function.a. f x( ) = 6x + 5

Example 2Give an equation for the inverse and tell whether it is

a function.a. f x( ) = 6x + 5

y = 6x + 5

Example 2Give an equation for the inverse and tell whether it is

a function.a. f x( ) = 6x + 5

y = 6x + 5

x = 6y + 5

Example 2Give an equation for the inverse and tell whether it is

a function.a. f x( ) = 6x + 5

y = 6x + 5

x = 6y + 5 −5 −5

Example 2Give an equation for the inverse and tell whether it is

a function.a. f x( ) = 6x + 5

y = 6x + 5

x = 6y + 5 −5 −5

x − 5 = 6y

Example 2Give an equation for the inverse and tell whether it is

a function.a. f x( ) = 6x + 5

y = 6x + 5

x = 6y + 5 −5 −5

x − 5 = 6y

y =

x − 56

Example 2Give an equation for the inverse and tell whether it is

a function.a. f x( ) = 6x + 5

y = 6x + 5

x = 6y + 5 −5 −5

x − 5 = 6y

y =

x − 56 or

Example 2Give an equation for the inverse and tell whether it is

a function.a. f x( ) = 6x + 5

y = 6x + 5

x = 6y + 5 −5 −5

x − 5 = 6y

y =

x − 56 or

y =

16

x −56

Example 2b.

y =

43x − 1

Example 2b.

y =

43x − 1

x =

43y − 1

Example 2b.

y =

43x − 1

x =

43y − 1

3y − 1 =

4x

Example 2b.

y =

43x − 1

x =

43y − 1

3y − 1 =

4x

3y =

4x+ 1

Example 2b.

y =

43x − 1

x =

43y − 1

3y − 1 =

4x

3y =

4x+ 1

Example 2b.

y =

43x − 1

x =

43y − 1

3y − 1 =

4x

3y =

4x+ 1

3y =

4x+

xx

Example 2b.

y =

43x − 1

x =

43y − 1

3y − 1 =

4x

3y =

4x+ 1

3y =

4x+

xx

Example 2b.

y =

43x − 1

x =

43y − 1

3y − 1 =

4x

3y =

4x+ 1

3y =

4x+

xx

3y =4 + x

x

Example 2b.

y =

43x − 1

x =

43y − 1

3y − 1 =

4x

3y =

4x+ 1

3y =

4x+

xx

3y =4 + x

x

y =

4 + x3x

Question:How do you verify that two functions are inverses of

each other?

Question:How do you verify that two functions are inverses of

each other?

Use the Inverse Function Theorem!

Question:How do you verify that two functions are inverses of

each other?

Use the Inverse Function Theorem!

The IFT says that two functions f and g are inverses of each other IFF f(g(x)) = x for all x in the domain of

g AND g(f(x)) = x for all x in the domain of f.

Example 3Verify that the functions in Example 2a are inverses of

each other.

Example 3Verify that the functions in Example 2a are inverses of

each other.

To do this, we have to show that f(g(x)) = x and g(f(x)) = x.

Example 3Verify that the functions in Example 2a are inverses of

each other.

To do this, we have to show that f(g(x)) = x and g(f(x)) = x.

Let’s calculate this together.

Example 4Explain why the functions f and g, with f(m) = m2 and

g(m) = m-2 are not inverses.

Example 4Explain why the functions f and g, with f(m) = m2 and

g(m) = m-2 are not inverses.

Calculate f(g(m)). If this composite does not give us a value of m, then we know they are not inverses. If it

does, then we have to check g(f(m)).

Homework

Homework

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