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### Inverse Functions

Bell Ringer

What do you think “inverse” means?

What is the rule for reflection over the line y = x?

Find the “sin” button on your calculator. What is written above it?

Thursday, March 13, 2014

The Basics

The inverse of a function is when you switch the x and y values.

The graph of inverse functions is a reflection over the line y = x.

Inverse function notation is f -1(x).

Ex: Switching x and y

Find the inverse of the given function:

f = {(1,2), (3,5), (-2,-5), (-7, 4)}

The inverse is:

f -1 = {(2,1), (5,3), (-5,-2), (4, -7)}

Finding the Inverse Algebraically

(See Graphic Organizer)

Switch to the y = notation from the f(x) =.

Exchange x and y in the problem and solve for y.

Rewrite as f -1(x).

Ex: Solving Algebraicallyf(x) = 3x2 - 8

- Switch notation
- Switch x & y and solve for y
- Add 8 to both sides
- Divide both sides by 3
- Take the square root
- Rewrite in function notation

- y = 3x2 – 8
- x = 3y2 – 8
- x + 8 = 3y2
- x + 8 = y2
3

√ (x + 8) = y

3

f -1(x) = ± √ (x + 8)

3

Ex: Solving Algebraically

- f(x) = √x
- y = √x
- x = √y
- x2 = y
- f -1 (x) = x2

- Copy
- Rewrite in y = form
- Switch x and y & solve for y
- Square both sides
- Write in function notation

Limits

If a function has a limited range, then the domain of its inverse is limited.

Ex: Limits

f(x) = √x

The range of this function is y ≥ 0

This means the inverse function has a limited domain of x ≥ 0

So, f -1(x) = x2, x ≥ 0

Exit Ticket

What are the 3 ways to determine the inverse of a function?

Which way do you find easiest? Explain.

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