Inverse functions are a critical concept in AP Precalculus, particularly in section 2.8. This answer key provides solutions for various inverse function problems, including finding inverses for quadratic, square root, and rational functions. Students preparing for the AP exam can use this resource to check their work and understand the domain and range of inverse functions. The key includes detailed explanations for each solution, making it a valuable study tool for mastering this topic.

Key Points

  • Includes solutions for finding inverses of quadratic and rational functions
  • Covers domain and range for each inverse function
  • Provides step-by-step explanations for each problem
  • Helps AP Precalculus students prepare for the May exam
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© The Algebros from FlippedMath.com
2.8 Inverse Functions
AP Precalculus
Find the inverse of each function and list the domain and range of
𝒇
𝟏
󰇛
𝒙
󰇜
.
1.
𝑓
󰇛
𝑥
󰇜
󰇛𝑥3󰇜
4
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
2.
𝑓
󰇛
𝑥
󰇜
𝑥6
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
3.
𝑓
󰇛
𝑥
󰇜
󰇛
𝑥1
󰇜
2 for 𝑥1
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
4.
𝑓
󰇛
𝑥
󰇜
√
𝑥2 3
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
5.
𝑓
󰇛
𝑥
󰇜
󰇛𝑥2󰇜
5 for 𝑥2
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
6.
𝑓
󰇛
𝑥
󰇜

Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
7.
𝑓
󰇛
𝑥
󰇜

󰇛
𝑥4
󰇜
1 for 𝑥4
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
8.
𝑓
󰇛
𝑥
󰇜

√
𝑥1 3
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
9.
𝑓
󰇛
𝑥
󰇜
2𝑥5
3𝑥4
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
2.8 Practice
© The Algebros from FlippedMath.com
Use the tables below to find the given values.
10.
𝒙
𝒇
󰇛
𝒙
󰇜
1 2
2 3
3 6
4 2
5 4
6 1
11.
𝒙
𝒇
󰇛
𝒙
󰇜
3 2
1 7
2 10
7 3
8 1
10 8
12.
𝒙
𝒇
󰇛
𝒙
󰇜
10 6
6 7
2 10
3 11
7 3
11 2
a.
𝑓
󰇛
1
󰇜
b. 𝑓
󰇛
6
󰇜
c.
𝑓

󰇛
1
󰇜
d.
𝑓

󰇛
4
󰇜
e. 𝑓
󰇛
2
󰇜
f.
𝑓

󰇛
6
󰇜
a.
𝑓
󰇛
2
󰇜
b. 𝑓
󰇛
10
󰇜
c.
𝑓

󰇛
7
󰇜
d.
𝑓

󰇛
3
󰇜
e. 𝑓
󰇛
7
󰇜
f.
𝑓

󰇛
2
󰇜
a.
𝑓
󰇛
10
󰇜
b. 𝑓
󰇛
3
󰇜
c.
𝑓

󰇛
7
󰇜
d.
𝑓

󰇛
6
󰇜
e. 𝑓
󰇛
7
󰇜
f.
𝑓

󰇛
2
󰇜
The graph of a function is given below. Identify if the function is invertible. Sketch the graph of the inverse
regardless of whether or not it is invertible.
13.
Is 𝑓
󰇛
𝑥
󰇜
invertible?
14.
Is 𝑔
󰇛
𝑥
󰇜
invertible?
15.
Is ℎ
󰇛
𝑥
󰇜
invertible?
Determine if the two functions are inverses of each other using composition.
16.
𝑓
󰇛
𝑥
󰇜
3𝑥5 and 𝑔
󰇛
𝑥
󰇜
𝑥
17.
𝑓
󰇛
𝑥
󰇜
√
3 𝑥
and 𝑔
󰇛
𝑥
󰇜
𝑥
3
𝑔󰇛𝑥󰇜
ℎ󰇛𝑥󰇜
𝑓
󰇛𝑥󰇜
© The Algebros from FlippedMath.com
18. The graph of the piecewise-linear function 𝑓 is shown in the figure. Let 𝑔
be the inverse function of 𝑓. What is the minimum value of 𝑔?
(A) 4
(B) 3
(C) 2
(D) 3
19. Mr. Brust is filling up his backdoor kiddie pool with the water hose. The amount of water, in gallons, in the pool 𝑡
minutes after he turns on the water can be modeled by 𝑃, an increasing function of time 𝑡. Which of the following
gives a verbal representation of the function 𝑃

, the inverse of 𝑃?
(A) 𝑃

is an increasing function of the amount of time after the water is turned on.
(B) 𝑃

is a decreasing function of the amount of time after the water is turned on.
(C) 𝑃

is an increasing function of the amount of water in the pool.
(D) 𝑃

is a decreasing function of the amount of water in the pool.
20.
𝑥 1 2 3 4 5
𝑓
󰇛
𝑥
󰇜
18 10 3 1 26
Calculator active. Let 𝑓 be an increasing function for 𝑥0. The table gives values of 𝑓
󰇛
𝑥
󰇜
at selected values of 𝑥. The
function 𝑔 is given by 𝑔
󰇛
𝑥
󰇜



a. The function ℎ is defined by ℎ
󰇛
𝑥
󰇜
󰇛
𝑔∘𝑓
󰇜󰇛
𝑥
󰇜
𝑔𝑓
󰇛
𝑥
󰇜
. Find the value of ℎ
󰇛
2
󰇜
as a decimal approximation or
indicate that it is not defined.
b. Find the value of 𝑓

󰇛
1
󰇜
, or indicate that it is not defined.
2.8 Test Prep
2.8 Inverse Functions
Graph of
𝑓
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End of Document
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FAQs

What types of functions are included in the inverse functions section?
The inverse functions section includes a variety of function types such as quadratic functions, square root functions, and rational functions. Each type presents unique challenges when finding inverses, and the answer key provides detailed solutions for each. Understanding these different functions is essential for AP Precalculus students, as they often appear in exam questions.

How does the answer key assist in understanding inverse functions?
The answer key assists students by providing clear, step-by-step solutions to each problem related to inverse functions. It explains the process of finding inverses, including how to switch the x and y variables and solve for y. Additionally, the key outlines the importance of determining the domain and range of the inverse functions, which is crucial for a complete understanding of the topic.

What is the significance of domain and range in inverse functions?
The domain and range of a function and its inverse are closely related. When finding the inverse of a function, the domain of the original function becomes the range of the inverse function, and vice versa. Understanding this relationship helps students grasp the concept of inverse functions more thoroughly, which is essential for success in AP Precalculus.

Are there practice problems included in the answer key?
While the primary focus of the answer key is to provide solutions, it is typically accompanied by practice problems in the corresponding section of the textbook. These problems allow students to apply what they have learned about inverse functions and test their understanding. The answer key serves as a valuable resource for checking their work and ensuring they grasp the concepts.

What topics are essential for mastering inverse functions in AP Precalculus?
Key topics for mastering inverse functions include understanding the definition of an inverse function, the process of switching x and y in equations, and the significance of domain and range. Additionally, students should be familiar with various types of functions such as linear, quadratic, and rational functions, as these often require different approaches when finding inverses. A solid grasp of these concepts is crucial for success in AP Precalculus.