Home / AP® Exam / AP® PreCalculus / AP® Precalculus

AP Precalculus -2.8 Inverse Functions- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -2.8 Inverse Functions- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -2.8 Inverse Functions- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

The function \( f \) is given by \( f(x) = 4 \cdot 2^{(x-3)} \). If the function \( g \) is the inverse of \( f \), which of the following could define \( g(x) \)?
(A) \( \log_8 x + 3 \)
(B) \( \log_2(4x) + 3 \)
(C) \( \log_2\left(\frac{x}{4}\right) + 3 \)
(D) \( \log_2\left(\frac{x-3}{4}\right) \)
▶️ Answer/Explanation
Detailed solution

To find the inverse, solve \( y = 4 \cdot 2^{(x-3)} \) for \( x \):
Divide by 4: \( \frac{y}{4} = 2^{(x-3)} \)
Take \(\log_2\): \( \log_2\left(\frac{y}{4}\right) = x – 3 \)
Thus \( x = \log_2\left(\frac{y}{4}\right) + 3 \)
Swap \( x \) and \( y \): \( g(x) = \log_2\left(\frac{x}{4}\right) + 3 \)
✅ Answer: (C)

Question 

The function \( f \) is defined by \( f(x) = \sqrt{4 – x^2} \) for \( -2 \leq x \leq 0 \). Which of the following expressions defines \( f^{-1}(x) \)?
(A) \( -\sqrt{4 – x^2} \) for \( -2 \leq x \leq 0 \)
(B) \( \sqrt{4 – x^2} \) for \( -2 \leq x \leq 0 \)
(C) \( -\sqrt{4 – x^2} \) for \( 0 \leq x \leq 2 \)
(D) \( \sqrt{4 – x^2} \) for \( 0 \leq x \leq 2 \)
▶️ Answer/Explanation
Detailed solution

Let \( y = \sqrt{4 – x^2} \) with \( -2 \leq x \leq 0 \) and \( 0 \leq y \leq 2 \).
Solve for \( x \): \( y^2 = 4 – x^2 \) ⇒ \( x^2 = 4 – y^2 \) ⇒ \( x = -\sqrt{4 – y^2} \) (negative because \( x \leq 0 \)).
Swap \( x \) and \( y \): \( f^{-1}(x) = -\sqrt{4 – x^2} \).
The domain of \( f^{-1} \) is the range of \( f \), which is \( 0 \leq x \leq 2 \).
✅ Answer: (C)

Question 

 
 
 
 
 
 
 
 
 
 
 
 
The graph of the function \( y = f(x) \) is given. Which of the following is the graph of \( y = f^{-1}(x) \)?

▶️ Answer/Explanation
Detailed solution

The graph of \( y = f^{-1}(x) \) is the reflection of the graph of \( y = f(x) \) across the line \( y = x \).
From the given figure, \( f \) appears to be a curve that increases and spans large \( y \)-values for small \( x \) changes.
The inverse should thus have large \( x \)-values for small \( y \) changes, resembling the shape in option (D) where the axes appear swapped.
✅ Answer: (D) 

Question 

The function \( f \) is defined by \( f(x) = 4x^2 + 3 \) for \( x \geq 0 \). Which of the following expressions defines the inverse function of \( f \)?
(A) \( f^{-1}(x) = \frac{x^2}{4} – 3 \) for \( x \geq 0 \)
(B) \( f^{-1}(x) = \sqrt{\frac{x}{4}} – 3 \) for \( x \geq 0 \)
(C) \( f^{-1}(x) = \sqrt{\frac{x-3}{4}} \) for \( x \geq 3 \)
(D) \( f^{-1}(x) = \frac{\sqrt{x-3}}{4} \) for \( x \geq 3 \)
▶️ Answer/Explanation
Detailed solution

Let \( y = 4x^2 + 3 \), \( x \geq 0 \), so \( y \geq 3 \).
Solve for \( x \): \( y – 3 = 4x^2 \) ⇒ \( x^2 = \frac{y-3}{4} \) ⇒ \( x = \sqrt{\frac{y-3}{4}} \) (positive root since \( x \geq 0 \)).
Swap \( x \) and \( y \): \( f^{-1}(x) = \sqrt{\frac{x-3}{4}} \) for \( x \geq 3 \).
✅ Answer: (C)

Scroll to Top