Inverse of a Function

Section 5.6 Beginning on Page 276

What is the Inverse of a Function?The inverse of a function is a generic equation to find the input of the original function when given the output [finding x when given y]. Inverse functions undo each other.

To find the inverse of a function we switch x and y and solve for y. We can then write a rule for the inverse function.

If we are given (or find) a set of coordinate pairs for a function, we can swap the values of x and the values of y and we will have a set of coordinate pairs for the inverse of the function.

You can verify if one function is the inverse of the other by composing the functions.

The inverse of a function might not also be a function.

If the graph of a function passes the horizontal line test, its inverse is also a function.

Writing a Formula for the Input of a Function

Example 1: Let (a) Solve for . (b) Find the input when the output is -7

−3−3

22

𝑦−32

=𝑥

𝑥=𝑦−32

𝑥=(−7)−32

𝑥=−102

𝑥=−5 The input is -5 when the output is -7

Inverse Functions

Inverse Functions

Finding the Inverse of a Linear Function

Example 2: Find the inverse of

**Set equal to . Switch the roles of and and solve for .

𝑦= 𝑓 (𝑥 )=3 𝑥−1𝑦=3 𝑥−1

𝑥+1=3 𝑦𝑥+13

=𝑦

𝑥=3 𝑦−1

𝑔 (𝑥 )= 𝑥+13

Or

𝑔 (𝑥 )=13𝑥+13

Inverses of Nonlinear Functions

Finding the Inverse of a Quadratic Function

Example 3: Find the inverse of . Then graph the function and its inverse.

𝑓 (𝑥 )=𝑥2

𝑦=𝑥2

𝑥=𝑦 2

√𝑥=√𝑦 2±√𝑥=𝑦𝑔 (𝑥 )=± √𝑥

Since the domain ofis restricted to nonnegative values of x, the range of the inverse has to be restricted to nonnegative values of y.

The inverse of is

The Horizontal Line Test

Finding the Inverse of a Cubic FunctionExample 4: Consider the function Determine whether the inverse ofis a function. Then find the inverse.

First, sketch the graph of the function and perform the horizontal line test.

Since no horizontal line intersects the graph more than once,the inverse of f is a function.

𝑦=2 𝑥3+1 𝑥=2 𝑦 3+1

𝑥−1=2 𝑦3

𝑥−12

=𝑦3

3√ 𝑥−12 =3√𝑦 3 3√ 𝑥−12 =𝑦 g (𝑥 )=3√ 𝑥−12

Finding the Inverse of a Radical Function

Example 5: Consider the function . Determine whether the inverse of f is a function. Then find the inverse.

𝑥=2√𝑦−3𝑥2=√ 𝑦−3

( 𝑥2 )2

=(√𝑦−3)2

𝑥2

4=𝑦−3

𝑥2

4+3=𝑦 𝑔 (𝑥 )= 1

4𝑥2+3

Verifying Functions are Inverses

𝑓 (𝑔 (𝑥 ) )= 𝑓 ( 𝑥+13 )¿3 (𝑥+1

3 )−1 ¿ 𝑥+1−1 ¿ 𝑥

𝑔 ( 𝑓 (𝑥 ) )=𝑔 (3 𝑥−1 )¿(3 𝑥−1 )+1

3¿3 𝑥3 ¿ 𝑥

Step 1: Find the inverse. Step 2: Evaluate the inverse when

𝑆=4𝜋𝑟 2𝑆4𝜋

=𝑟 2

√ 𝑆4𝜋

=𝑟

𝑟=√ 𝑆4𝜋

𝑟=√ 100𝜋4𝜋

𝑟=√25𝑟=5The radius of the sphere is 5 ft.

Monitoring ProgressSolve for . Then find the input(s) when the output is 2.

1) 2) 3)

Find the inverse of the function. Then graph the function and its inverse.

4) 5) 6)

𝑦+2=𝑥 ;4 ±√ 𝑦2 =𝑥 ;±1 3√3− 𝑦=𝑥 ;1

Monitoring Progress

Monitoring Progress

10) Yes11) No