Notes: (Topic 2.

8) Inverse Functions

An inverse relation will “undo” a given relation. Every inverse relation can be found by ______________ each x and
y value.

In some situations, this process is intuitive, but other times this may not be as obvious. In these cases, it is important that
we understand the ___________________ of inverse relations as they will help us tackle a variety of problems.

Let’s look at this numerically (tables), graphically (pictures) and analytically (equations):

Numerical (Tables)

Example 1: Find the inverse relation of the given table.

𝑥 1 3 4 6 𝑥

𝑦 −1 2 0 2 𝑦

Note: The original table ______ a function because each _________ has exactly one _________. However, the
inverse is ______ a function because _____________________________________________________.

Graphical (Pictures)

Example 2: Sketch the inverse relation of the given graph.

Steps to Sketch an Inverse Graph

(Linear Pieces)
1. List the key points in a table
2. Create a new table for the inverse by switching
the x and y values.
3. Plot the new points for the inverse and sketch
the inverse graph.

Example 2B: Is the original relation a function? Is the inverse relation a function?

Graphical Property of Inverses: The graphs of inverses are ___________________ over the line _________.

Notes: Inverse Functions Topic 2.8 Created by Bryan Passwater

Analytical (Equations)

Example 3: Find the inverse for 𝑓(𝑥 ) = 3𝑥 − 7 (Reminder: 𝑓(𝑥 ) = 𝑦)

𝐒𝐭𝐞𝐩𝐬 𝐭𝐨 𝐅𝐢𝐧𝐝 𝐚𝐧 𝐈𝐧𝐯𝐞𝐫𝐬𝐞 𝐄𝐪𝐮𝐚𝐭𝐢𝐨𝐧

1. 𝐒𝐰𝐢𝐭𝐜𝐡 the 𝑥 and 𝑦 values.
2. 𝐒𝐨𝐥𝐯𝐞 for 𝑦 (Get 𝑦 by itself!)
3. If original equation was in 𝑓(𝑥 ) = form,
Notation: The inverse function of 𝑓 (𝑥 ) is written as ________. write the inverse equation as 𝑓 −1 (𝑥 ) =

𝐅𝐢𝐧𝐝𝐢𝐧𝐠 𝐈𝐧𝐯𝐞𝐫𝐬𝐞 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬 𝐨𝐟 𝐑𝐚𝐭𝐢𝐨𝐧𝐚𝐥 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬 (𝐰𝐢𝐭𝐡 𝐦𝐮𝐥𝐭𝐢𝐩𝐥𝐞 𝒙′𝐬)

𝐒𝐭𝐞𝐩 𝟏 Switch the 𝑥 and 𝑦 values.

Multiply/Distribute both sides by the denominator of the rational expression to eliminate

𝐒𝐭𝐞𝐩 𝟐
the fraction.
Move all terms that include the variable 𝑦 to the left side of the equation and move all
𝐒𝐭𝐞𝐩 𝟑
terms that do not include the variable 𝑦 to the right side of the equation.

𝐒𝐭𝐞𝐩 𝟒 Factor out an 𝑦 from the terms on the left side of the equation.

𝐒𝐭𝐞𝐩 𝟓 Divide both sides by the terms remaining on the left side after 𝑦 was factored out.

𝐒𝐭𝐞𝐩 𝟔 Rewrite the equation with proper inverse notation (𝑓 −1 (𝑥 ))

𝐄𝐱𝐚𝐦𝐩𝐥𝐞 𝟒: Find the inverse functions for each of the following.

𝑥−2 2𝑥 + 1
a) 𝑓 (𝑥 ) = b) 𝑔(𝑥 ) =
𝑥+3 𝑥−3

Notes: Inverse Functions Topic 2.8 Created by Bryan Passwater

𝐌𝐨𝐫𝐞 𝐨𝐧 𝐆𝐫𝐚𝐩𝐡𝐢𝐧𝐠 𝐈𝐧𝐯𝐞𝐫𝐬𝐞𝐬
We know that to find an inverse, we simply switch the ____ and ____ values.

Graphically, this means that a function and its inverse will be reflections over the line _________.

If a graph is not made up of simply line segments, sketching the inverse can be challenging. To do this, we use the
fact that a function and its inverse are reflections over the line 𝑦 = 𝑥 to help us sketch the inverse graph.

𝐒𝐭𝐞𝐩𝐬 𝐭𝐨 𝐒𝐤𝐞𝐭𝐜𝐡 𝐚𝐧 𝐈𝐧𝐯𝐞𝐫𝐬𝐞 𝐆𝐫𝐚𝐩𝐡 (Nonlinear pieces)

1. Sketch the line ______ on the graph.

2. Mark any points from the original graph that are already ____ the line. These points will stay the same!

3. Select a few additional points on the original graph and find their inverse points.

• You can find the inverse points by switching the 𝑥 and 𝑦 values, 𝐎𝐑
• Graphically reflecting each point over the line 𝑦 = 𝑥 by drawing a line ________________________
from the point to the line 𝑦 = 𝑥 and extending it an equal distance on the other side of the line.
4. Sketch the inverse graph by connecting the new points in a similar pattern to the original function.

𝐄𝐱𝐚𝐦𝐩𝐥𝐞 𝟓: Sketch the inverse of the functions below.

a) b)

𝐖𝐡𝐚𝐭 𝐝𝐨 𝐈𝐧𝐯𝐞𝐫𝐬𝐞 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬 𝐝𝐨?

If you plug a number into a function, it generally will output a new (different) number. For example, if we plug in
𝑥 = 2 into the function 𝑓(𝑥 ) = 3𝑥 − 1, we get ________.

Now, If we plug that 𝐚𝐧𝐬𝐰𝐞𝐫 into the inverse function, the answer should be _____.

This is because an inverse function essentially will “undo” a function. So, if we plug _____ into a function, and
then plug the output into the inverse function, we should end up with _____ again.

Notes: Inverse Functions Topic 2.8 Created by Bryan Passwater

 𝐈𝐧𝐯𝐞𝐫𝐬𝐞 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬
Two functions 𝑓 (𝑥 ) and 𝑔(𝑥) are inverses if and only if 𝑓(𝑔(𝑥 )) = 𝑥 and 𝑔(𝑓 (𝑥 )) = 𝑥.

To show that two functions are inverses, we must show that the ________________________ 𝐁𝐎𝐓𝐇 equal _____.

𝐄𝐱𝐚𝐦𝐩𝐥𝐞 𝟔: Let 𝑓(𝑥 ) and 𝑔(𝑥 ) be the functions below. Determine if 𝑓(𝑥 ) and 𝑔(𝑥 ) are inverses.
1 3
𝑓(𝑥 ) = 2𝑥 − 3 𝑔 (𝑥 ) = 𝑥+
2 2

𝐒𝐭𝐞𝐩 𝟏: Find 𝑓(𝑔(𝑥 ))

𝐒𝐭𝐞𝐩 𝟐: Find 𝑔(𝑓(𝑥 ))

Note: Sometimes, we must use a restricted domain to ensure two functions are inverses.

𝐄𝐱𝐚𝐦𝐩𝐥𝐞 𝟕: Show that ℎ(𝑥 ) and 𝑘(𝑥 ) below are inverses where 𝑥 ≥ 10.

ℎ(𝑥 ) = 𝑥 2 + 10 𝑘 (𝑥 ) = √𝑥 − 10

𝐄𝐱𝐚𝐦𝐩𝐥𝐞 𝟖: Show that 𝑛(𝑥 ) and 𝑝(𝑥 ) below are inverses where 𝑥 ≠ 0 and 𝑥 ≠ 4.
6 6
𝑛 (𝑥 ) = 𝑝 (𝑥 ) = +4
𝑥−4 𝑥

Notes: Inverse Functions Topic 2.8 Created by Bryan Passwater

Notes: (Topic 2.8) Inverse Functions – Part II

Inverse Functions
If 𝑓 and 𝑔 are inverse functions, then…

1. 𝑔(𝑥) = 𝑓 −1 (𝑥)
2. If (𝑥, 𝑦) is a point on the graph of 𝑓(𝑥), then (𝑦, 𝑥) is a point on the graph of 𝑔(𝑥).
3. With inverse functions, all of the 𝑥 and 𝑦 values are “switched”, so the graphical behaviors in terms of 𝑥 and 𝑦
will also be switched. For example, the domain of 𝒇 is the range of 𝒇−𝟏 .
4. A continuous function will only have an inverse function if it is strictly increasing or strictly decreasing. If a
function changes from increasing to decreasing (or vice versa), it will not pass the horizontal line test and its inverse
relation will not pass the vertical line test as a result.

𝑥 −3 −2 0 1 4 6

𝑓(𝑥) 6 3 1 −1 −3 −7

Example 1: Let 𝑓 be a continuous function with selected values in the table below. Let 𝑔 be the inverse of 𝑓, such that
𝑔(𝑥) = 𝑓 −1 (𝑥). Find the following values if possible.
a) 𝑓(𝑓(0)) b) 𝑔(−3) c) 𝑔(6)

d) 𝑔(𝑔(−1)) e) (𝑓 −1 ∘ 𝑓)(−2) f) 𝑓 −1 (−3)

𝐆𝐫𝐚𝐩𝐡 𝐨𝐟 𝒌
Example 2: The function 𝑘 is defined over the interval −4 ≤ 𝑥 ≤ 11 as shown above. Let 𝑘 −1 represent the inverse of 𝑘.
a) What is the minimum value of 𝑘(𝑥)? What is the minimum value of 𝑘 −1 (𝑥)?

b) Find 𝑘 −1 (6) and 𝑘 −1 (4).

Notes: Inverse Functions Topic 2.8 Created by Bryan Passwater

 𝐆𝐫𝐚𝐩𝐡 𝐨𝐟 𝒇
Example 3: The function 𝑓 is defined over the interval −2 ≤ 𝑥 ≤ 8 as shown above. Let 𝑓 −1 represent the inverse of 𝑓.

a) What is the maximum value of 𝑓 −1 (𝑥)?

b) Find 𝑓 −1 (3) and 𝑓 −1 (1).

c) What is the domain of 𝑓 −1 ?

𝑥 −5 −1 0 2 5 6

ℎ(𝑥) −3 0 3 5 8 10

𝐆𝐫𝐚𝐩𝐡 𝐨𝐟 𝒈
Example 4: The function 𝑔 is defined over the interval −3 ≤ 𝑥 ≤ 10 as shown above. Let 𝑔−1 represent the inverse of 𝑔.
Values of the increasing function ℎ are given in the table above for selected values of 𝑥. Find the following, if possible.

a) 𝑔(ℎ(6)) b) 𝑔−1 (ℎ(0))

c) ℎ−1 (𝑔(8)) d) ℎ−1 (𝑔−1 (−1))

Notes: Inverse Functions Topic 2.8 Created by Bryan Passwater