Section 3.

7: Inverse Functions

• Verify inverse functions.

• Determine the domain and range of an inverse function and restrict the domain of a function to
make it one-to-one.
• Find or evaluate the inverse of a function.
• Use the graph of a one-to-one function to graph its inverse function on the same axes.

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Combining Functions Using Algebraic Operations

When we think about the inverse of a function, we want to know if the function works in reverse,
meaning, if inputs become outputs and outputs become inputs, is the result still a function? See this
image:

Example 1:

i. If we reverse the inputs and outputs of the function 𝑓(𝑥) = {(−2,6), (0, −3), (4,10)}, the result is
{(6, −2), (−3,0), (10,4)}.

Does this result represent a function? Why or why not?

ii. Let 𝑔(𝑥) = {(−7, −1), (0,5), (1,0), (4,5)}. If we switch the inputs and outputs, we get
{(−1, −7), (5,0), (0,1), (5,4)}.

Does this result represent a function? Why or why not?

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iii. Compare 𝑓(𝑥) = {(−2,6), (0, −3), (4,10)}, and 𝑔(𝑥) = {(−7, −1), (0,5), (1,0), (4,5)}.

What is the difference between the two functions that causes 𝑓(𝑥) to have an inverse function but 𝑔(𝑥)
does not?

Think-Pair-Share 1:

Compare 𝑓(𝑥) = {(−2,6), (0, −3), (4,10)}, and 𝑔(𝑥) = {(−7, −1), (0,5), (1,0), (4,5)}.

Write the domain and range of 𝑓(𝑥):

Write the domain and range of 𝑔(𝑥):

What is the difference between the two functions that causes 𝑓(𝑥) have an inverse function but 𝑔(𝑥)
does not?

A function must be one-to-one to have an inverse function. In other words, only one-to-one functions
have inverses.

Recall that a function is one-to-one if there is no repeated output, that the graph passes the horizontal
line test.

In the example above, you notice that 𝑓(𝑥) = {(−2,6), (0, −3), (4,10)} is one-to-one.

Thus, 𝑓 −1 (𝑥) exists.

However, 𝑔(𝑥) = {(−7, −1), (0,5), (1,0), (4,5)} is NOT one-to-one, so there is no inverse function for it.

Important Notes:
𝟏
1. The inverse of a one-to-one function 𝒇(𝒙), is 𝒇−𝟏 (𝒙). Note that 𝒇−𝟏 (𝒙) ≠ 𝒇(𝒙).

2. 𝒇−𝟏 (𝒙) is found by switching inputs values of 𝒇(𝒙) with its output values.

3. For a one-to-one function 𝒇(𝒙) and its inverse 𝒇−𝟏 (𝒙), domain and range switch, meaning:

• The set that shows the domain of 𝑓(𝑥) is equal to the set that
shows the range of 𝑓 −1 (𝑥), and vice-versa.
• Also, the set that shows the range of 𝑓(𝑥) is equal to the set that
shows the domain of 𝑓 −1 (𝑥), and vice-versa.

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Example 2:

i. Suppose that the domain of the one-to-one function 𝑓(𝑥) is [−2, ∞) and its range is (−∞, 6).

What is the domain of 𝑓 −1 (𝑥)?

What is the range of 𝑓 −1 (𝑥)?

ii. See the graph of one-to-one function 𝑓(𝑥) below. Find the domain and range of the function and
based on that determine the domain and range of 𝑓 −1 (𝑥).

Domain of 𝑓(𝑥): Domain of 𝑓 −1 (𝑥):

Range of 𝑓(𝑥): Range of 𝑓 −1 (𝑥):

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4. The graphs of 𝒇(𝒙) and 𝒇−𝟏 (𝒙) are reflections of each other over the line 𝒚 = 𝒙.

Example 3:

a) Verify that the function 𝑓(𝑥) = (−7, −2), (−1,0), (2, −3), (6,1) is one-to-one by finding the domain
and the range of 𝑓(𝑥) and 𝑓 −1 (𝑥).

Domain of 𝑓(𝑥): Domain of 𝑓 −1 (𝑥):

Range of 𝑓(𝑥): Range of 𝑓 −1 (𝑥):

b) Graphs of 𝑓(𝑥), 𝑓 −1 (𝑥) and 𝑦 = 𝑥 are shown below. Notice the reflection of the points over the line
𝑦 = 𝑥:

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Think-Pair-Share 3:

The graph below shows 𝑔(𝑥) and the line 𝑦 = 𝑥.

a) Is 𝑔(𝑥) one-to-one?

b) Sketch the graph of 𝑔−1 (𝑥) in the same grid below.

5. The composition of a one-to-one function 𝒇(𝒙) with its inverse 𝒇−𝟏 (𝒙) always results in the identity
function:

(𝑓 ∘ 𝑓 −1 )(𝑥) = (𝑓 −1 ∘ 𝑓)(𝑥) = 𝑥

Or:

𝑓(𝑓 −1 (𝑥)) = 𝑓 −1 (𝑓(𝑥)) = 𝑥

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Example 4:
3
i) If 𝑓(𝑥) = 𝑥 3 + 2 and 𝑔(𝑥) = √𝑥 − 2, is 𝑔(𝑥) = 𝑓 −1 (𝑥)?

1
ii) If 𝑓(𝑥) = 4𝑥 3 and 𝑔(𝑥) = 4 𝑥, is 𝑔(𝑥) = 𝑓 −1 (𝑥)?

Think-Pair-Share 4:
1 1
If 𝑓(𝑥) = 𝑥+3 and 𝑔(𝑥) = 𝑥 − 3, 𝑔(𝑥) = 𝑓 −1 (𝑥)?

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How to find the inverse of a function represented by a formula

1. Make sure the function is one-to-one. Then replace 𝑓(𝑥) with 𝑦.

2. Interchange 𝑥 and 𝑦.
3. Solve for 𝑦.
4. Replace 𝑦 with 𝑓 −1 (𝑥).

Note: You can switch the order and first solve for 𝑥 and then interchange 𝑥 and 𝑦.

Example 5:
2
a) Find the inverse of the function 𝑓(𝑥) = 𝑥−1 + 6.

b) Find the inverse of 𝑔(𝑥) = 3 + √𝑥 − 1. Give the domain and range for 𝑔(𝑥) and 𝑔−1 (𝑥) using
interval notation.

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Think-Pair-Share 5:

The function ℎ(𝑥) = 𝑥 2 − 6 is not one-to-one, but if we restrict its domain to [0, ∞) then it becomes
one-to-one and has an inverse. (Left image is h(𝑥) = 𝑥 2 − 6, right image is h(𝑥) = 𝑥 2 − 6, [0, ∞))

i. What is the range of ℎ(𝑥)?

ii. Find the inverse of ℎ(𝑥).

iii. Determine the domain and range of ℎ−1 (𝑥).

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