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)

Question

The graph of the piecewise-linear function $ f $ is shown in the figure. Let $ g $ be the inverse function of $ f $. What is the maximum value of $ g $?

(A) $ \frac{1}{7} $
(B) $ \frac{1}{5} $
(C) 5
(D) 7

▶️ Answer/Explanation

Detailed solution

From the graph, $ f $ is increasing and its maximum output is $ y = 7 $ at $ x = 5 $ (endpoint).
Since $ g $ is the inverse of $ f $, the domain of $ g $ is the range of $ f $, and the range of $ g $ is the domain of $ f $.
The range of $ g $ is $ [0,5] $, so the maximum value of $ g $ is 5, which occurs when the input to $ g $ is 7 because $ f(5) = 7 $ ⇒ $ g(7) = 5 $.
✅ Answer: (C)

Question

Consider the function $f(x)$ shown in the graph. Which of these graphs represents the inverse of $f(x)$?

▶️ Answer/Explanation

Detailed solution

The inverse of a function $f(x)$ is found by reflecting its graph across the line $y = x$.
In the original graph, the curve passes through the origin $(0,0)$.
The original function has a slow horizontal growth in the first quadrant and a steep drop in the third quadrant.
When reflected across $y = x$, the horizontal behavior becomes vertical and the vertical becomes horizontal.
This results in a graph that is steep in the first quadrant and flattens out in the third quadrant.
Comparing the choices, Option (a) correctly represents this reflection.
Therefore, the correct representation of the inverse $f^{-1}(x)$ is a.

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