AP PreCalculus Practice Quiz: Inverse Functions

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 16 questions to check your progress.

Question 1 of 16

The function f contains the ordered pair (3, 8). If f is an invertible function, which of the following ordered pairs must be on the graph of its inverse function, f⁻¹?

A)(8, 3)B)(3, 8)C)(-3, -8)D)(-8, -3)

All Questions (16)

The function f contains the ordered pair (3, 8). If f is an invertible function, which of the following ordered pairs must be on the graph of its inverse function, f⁻¹?

A) (8, 3)

B) (3, 8)

C) (-3, -8)

D) (-8, -3)

Correct Answer: A

Based on the content, the input-output pairs of the inverse of a function are determined by reversing the input-output pairs of the original function. If (3, 8) is a pair for f, then (8, 3) must be a pair for f⁻¹.

Let f be an invertible function. What is the result of the composition f⁻¹(f(x))?

A) 1

B) 0

C) x

D) f(x)

Correct Answer: C

The provided content states that the composition of a function, f, and its inverse function, f⁻¹, is the identity function; that is, f(f⁻¹(x)) = f⁻¹(f(x)) = x.

The graph of the inverse of a function y = f(x) can be found by reflecting the graph of f over which line?

A) The x-axis

B) The y-axis

C) The line y = -x

D) The line y = x

Correct Answer: D

The content specifies that the inverse of the graph of the function y = f(x) can be found by reflecting the graph of the function over the graph of the identity function h(x) = x.

An invertible function f has a domain of [0, 10] and a range of [-2, 5]. What is the domain of the inverse function, f⁻¹?

A) [0, 10]

B) [-2, 5]

C) [-10, 0]

D) [-5, 2]

Correct Answer: B

According to the provided content, on a function's invertible domain, the function's range becomes the inverse function's domain. Therefore, the domain of f⁻¹ is the range of f, which is [-2, 5].

What condition must be met for a function f to be invertible on a specified domain?

A) The function must be continuous on the domain.

B) The function must have a positive slope on the domain.

C) Each output value of f must be mapped from a unique input value.

D) The domain and range of the function must be the same.

Correct Answer: C

The content explicitly states that on a specified domain, a function, f, has an inverse function, or is invertible, if each output value of f is mapped from a unique input value.

The function f is defined by f(x) = x - 7. Which of the following is the inverse function, f⁻¹(x)?

A) f⁻¹(x) = x + 7

B) f⁻¹(x) = 7 - x

C) f⁻¹(x) = 7x

D) f⁻¹(x) = x - 7

Correct Answer: A

To find the inverse, we must determine the inverse operations to reverse the mapping. The function f subtracts 7 from the input. The inverse operation is to add 7, so f⁻¹(x) = x + 7.

Let f be an invertible function, and let g be its inverse, g(x) = f⁻¹(x). If f(5) = 12, what is the value of g(12)?

A) 5

B) 12

C) 1/5

D) 1/12

Correct Answer: A

Since f and g are inverses, their input-output pairs are reversed. If f maps the input 5 to the output 12, then its inverse g must map the input 12 back to the output 5.

Let f be an invertible function. Which of the following statements is equivalent to the expression f(f⁻¹(5)) = 5?

A) The function f has a value of 5 when x=0.

B) The composition of a function and its inverse yields the input value.

C) The graph of f has a point at (5, 5).

D) The domain of f is all real numbers.

Correct Answer: B

This is a specific application of the general rule provided in the content: the composition of a function, f, and its inverse function, f⁻¹, is the identity function. Here, the input is 5, and the final output is 5.

An invertible function g has a range of (–∞, 0]. What is the domain of its inverse, g⁻¹?

A) [0, ∞)

B) (–∞, 0]

C) (–∞, ∞)

D) The domain cannot be determined.

Correct Answer: B

The content states that a function's range is the inverse function's domain. Therefore, the domain of g⁻¹ is the same as the range of g, which is (–∞, 0].

The function f(x) = 2x + 3 is invertible for all real numbers. How is its inverse, f⁻¹(x), found using the principle of inverse operations?

A) First add 3, then multiply by 2.

B) First divide by 2, then subtract 3.

C) First subtract 3, then divide by 2.

D) First multiply by 1/2, then add 3.

Correct Answer: C

The function f(x) first multiplies the input by 2, then adds 3. To reverse this mapping, we must apply the inverse operations in reverse order: first subtract 3, then divide by 2. This gives f⁻¹(x) = (x-3)/2.

The function h(x) = x² is not invertible on the domain (–∞, ∞) because it maps multiple inputs to the same output (e.g., h(-2)=4 and h(2)=4). On which of the following domains is h(x) invertible?

A) [-5, 5]

B) [0, ∞)

C) (-1, 1)

D) (–∞, ∞)

Correct Answer: B

For a function to be invertible, each output value must be mapped from a unique input value. On the domain [0, ∞), every non-negative x value produces a unique output, so the function is invertible on this restricted domain.

If the point (c, d) lies on the graph of an invertible function f, which of the following points must lie on the graph of f⁻¹?

A) (c, d)

B) (-c, -d)

C) (d, c)

D) (1/c, 1/d)

Correct Answer: C

The relationship between a function and its inverse involves swapping the roles of the x- and y-coordinates. This is because the input-output pairs are reversed, and the graph is reflected over y=x.

Let f be an invertible function such that f(10) = 4. Which of the following statements MUST be true?

A) f⁻¹(10) = 4

B) f(4) = 10

C) f(f⁻¹(4)) = 4

D) The graph of f passes through the origin.

Correct Answer: C

The statement f(f⁻¹(x)) = x is always true for any value x in the domain of f⁻¹. Since f(10)=4, we know 4 is in the range of f, which means 4 is in the domain of f⁻¹. Therefore, f(f⁻¹(4)) must equal 4. Option A is incorrect because f⁻¹(4) = 10. Option B does not necessarily follow.

The process of finding the inverse of a function y = f(x) graphically involves reversing the roles of the x- and y-axes. This is geometrically equivalent to what transformation?

A) A 90-degree rotation about the origin.

B) A reflection over the x-axis.

C) A reflection over the line y = x.

D) A translation horizontally and vertically.

Correct Answer: C

The provided content explicitly states that the inverse of the graph of y = f(x) is found by reflecting the graph of the function over the graph of the identity function h(x) = x.

A table of values for an invertible function f is given. What is the value of f⁻¹(5)? x | f(x) --|--- 1 | 3 2 | 5 3 | 7 4 | 9

A) 1

B) 2

C) 3

D) 7

Correct Answer: B

Finding f⁻¹(5) means finding the input to the inverse function that gives an output of 5. This is equivalent to finding the input to the original function f that gives an output of 5. From the table, f(2) = 5. Therefore, f⁻¹(5) = 2.

If f and g are inverse functions, and the range of f is the set of all real numbers (–∞, ∞), what can be concluded about the function g?

A) The domain of g is (–∞, ∞).

B) The range of g is (–∞, ∞).

C) The graph of g is a reflection of f over the y-axis.

D) The composition g(f(x)) is undefined.

Correct Answer: A

The content states that the range of a function is the domain of its inverse. Since g is the inverse of f, the domain of g must be equal to the range of f. Therefore, the domain of g is (–∞, ∞). We cannot conclude anything about the range of g without knowing the domain of f.