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.

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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.

Question

Mr. Prashar 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 $P^{-1}$?

(A) $P^{-1}$ is an increasing function of the amount of time after the water is turned on.

(B) $P^{-1}$ is a decreasing function of the amount of time after the water is turned on.

(D) $P^{-1}$ is a decreasing function of the amount of water in the pool.

▶️ Answer/Explanation

Detailed solution

The function $P$ takes time as the input and outputs the amount of water.
Therefore, the inverse function $P^{-1}$ must swap these roles: it takes the amount of water as the input and outputs time.
Since $P$ is an increasing function (more time equals more water), its inverse must also be increasing.
Logically, to reach a greater amount of water (input for $P^{-1}$), a greater amount of time (output for $P^{-1}$) is required.
Thus, $P^{-1}$ is an increasing function of the amount of water in the pool.
This description matches option (C).

Question

A logarithmic function $ g $ is created by transforming the function $ f(x) = \log_5(x) $ such that $ g(x) = \frac{1}{2}f(x+3) + 2 $. What is the value of $ g^{-1}(3) $?

(A) $ 55 $

(B) $ \sqrt{7} $

(D) $ 22 $

▶️ Answer/Explanation

Detailed solution

To find the value of $ g^{-1}(3) $, we need to find the value of $ x $ such that $ g(x) = 3 $.
First, substitute $ f(x) $ into the given equation for $ g(x) $:
$ g(x) = \frac{1}{2}\log_5(x+3) + 2 $
Set the equation equal to 3 and solve for $ x $:
$ \frac{1}{2}\log_5(x+3) + 2 = 3 $
Subtract 2 from both sides: $ \frac{1}{2}\log_5(x+3) = 1 $
Multiply both sides by 2: $ \log_5(x+3) = 2 $
Convert the logarithmic equation to exponential form: $ x + 3 = 5^2 $
Solve for $ x $: $ x = 25 – 3 = 22 $.
Therefore, the correct option is (D).

Question

The table gives values for the invertible functions $h$ and $k$ at selected values of $x$. What is the value of $h^{-1}(k(3))$ ?

(A) 1

(B) 2

(D) 12

▶️ Answer/Explanation

Detailed solution

Step 1: To evaluate the composite function $h^{-1}(k(3))$, we must first determine the value of the inner function, $k(3)$.

Step 2: Looking at the provided table, find the column where $x = 3$. In the row for $k(x)$, the corresponding value is $4$. Therefore, $k(3) = 4$.

Step 3: Now, substitute this result back into the expression. We need to find $h^{-1}(4)$.

Step 4: The inverse function $h^{-1}(4)$ asks for the input value $x$ such that $h(x) = 4$.

Step 5: Inspect the row for $h(x)$ to find the value $4$. This value appears in the column where $x = 1$.

Step 6: Since $h(1) = 4$, it follows that $h^{-1}(4) = 1$.

Final Answer: The correct option is (A).

Question

The table gives values for the invertible function $y = f(x)$ for selected values of $x$. Which of the following input-output pairs describes $f^{-1}$?

(A)

(B)

(C)

(D)

▶️ Answer/Explanation

Detailed solution

The inverse function $f^{-1}$ swaps the inputs and outputs of the original function $f$.
From the table, the original pairs $(x, y)$ are $(-2, 1)$, $(1, 6)$, $(2, 2)$, and $(5, -3)$.
The inverse pairs $(y, x)$ must be $(1, -2)$, $(6, 1)$, $(2, 2)$, and $(-3, 5)$.
Sorting these by input value, we get: $(-3, 5)$, $(1, -2)$, $(2, 2)$, and $(6, 1)$.
Option (B) lists these input values $\{-3, 1, 2, 6\}$ with corresponding outputs $\{5, -2, 2, 1\}$.
Therefore, (B) is the correct description of the inverse function $f^{-1}$.

Question

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

(A)

(B)

(C)

(D)

▶️ Answer/Explanation

Detailed solution

The correct option is (D).
Identify key points on the graph of $y = f(x)$, such as $(-2, -10)$, $(0, 2)$, and $(2, 10)$.
For the inverse function $y = f^{-1}(x)$, the $x$ and $y$ coordinates are swapped.
The corresponding points on $f^{-1}(x)$ must be $(-10, -2)$, $(2, 0)$, and $(10, 2)$.
Graphically, $f^{-1}(x)$ is the reflection of $f(x)$ across the line $y = x$.
Option (D) is the only graph that contains these swapped points and reflects the original shape.
Notice the $x$-axis scale in (D) extends to $\pm 20$, matching the original $y$-axis range.

Question

The function $g$ is given by $g(x) = \frac{4x+6}{5}$. Which of the following defines $g^{-1}(x)$?

(A) $\frac{5}{4x+6}$
(B) $\frac{5x+6}{4}$
(C) $\frac{5x}{4} – 6$
(D) $\frac{5x-6}{4}$

▶️ Answer/Explanation

Detailed solution

Set the function $y = \frac{4x+6}{5}$.
Swap the variables $x$ and $y$ to get $x = \frac{4y+6}{5}$.
Multiply both sides by $5$ to obtain $5x = 4y + 6$.
Subtract $6$ from both sides: $5x – 6 = 4y$.
Divide by $4$ to solve for $y$: $y = \frac{5x-6}{4}$.
Therefore, $g^{-1}(x) = \frac{5x-6}{4}$, which matches option (D).

Question

A water tank is leaking water from a crack in its base. The amount of water, in hundreds of gallons, remaining in the tank $t$ hours after the crack formed can be modeled by $W$, a decreasing function of time $t$. Which of the following gives a verbal representation of the function $W^{-1}$, the inverse function of $W$?

(A) $W^{-1}$ is an increasing function of the amount of time after the crack formed.

(B) $W^{-1}$ is a decreasing function of the amount of time after the crack formed.

(D) $W^{-1}$ is a decreasing function of the amount of water in the tank.

▶️ Answer/Explanation

Detailed solution

The original function $W$ maps time $t$ to the amount of water $w$.
The inverse function $W^{-1}$ maps the amount of water $w$ back to time $t$.
Since $W$ is a decreasing function, as $t$ increases, $w$ decreases.
For any monotonic function, the inverse function $W^{-1}$ shares the same monotonicity as the original function.
Therefore, $W^{-1}$ must also be a decreasing function.
As the input of $W^{-1}$ is the “amount of water,” the correct description is that it is a decreasing function of the amount of water.
Thus, the correct option is (D).

Question

The table gives values of the function $f$ for selected values of $x$. Which of the following is a verbal representation of $f^{-1}(x)$, the inverse function of $f$?

(A) $f^{-1}(x)$ is logarithmic with input values increasing by $1$ every time output values double.

(B) $f^{-1}(x)$ is logarithmic with output values increasing by $1$ every time input values double.

(D) $f^{-1}(x)$ is exponential with output values increasing by $1$ every time input values double.

▶️ Answer/Explanation

Detailed solution

The original function $f$ is exponential because as $x$ increases by $1$, $f(x)$ doubles ($2, 4, 8, 16$).
The inverse of an exponential function is a logarithmic function, ruling out (C) and (D).
For the inverse function $f^{-1}(x)$, the inputs and outputs of $f$ are swapped.
The inputs of $f^{-1}(x)$ are the values $\{2, 4, 8, 16\}$ and the outputs are $\{1, 2, 3, 4\}$.
As the input values of $f^{-1}(x)$ double, the output values increase by $1$.
This matches the description provided in option (B).
Therefore, $f^{-1}(x)$ is logarithmic with output values increasing by $1$ every time input values double.

Question

The graph of the exponential function $f$ is given. Which of the following could be a table of values for the inverse function of $f$?

(A)

(B)

(C)

(D)

▶️ Answer/Explanation

Detailed solution

The graph shows $f$ passes through $(0, 1)$, $(1, 2)$, and $(2, 4)$.
This indicates the exponential function is defined by $f(x) = 2^x$.
An inverse function $f^{-1}(x)$ swaps input and output: $(x, y) \rightarrow (y, x)$.
Since $f(2) = 4$, the inverse must contain the point $(4, 2)$.
Since $f(5) = 2^5 = 32$, the inverse must contain the point $(32, 5)$.
Table (D) is the only one correctly matching these coordinates.
Therefore, (D) is the correct table for the inverse function.

Question

In a certain town, the population in the year $2000$ was about $30,000$. The population grows at a rate of $2.3\%$ per year, and time is measured in years since $2000$. Which of the following functions gives output values, in years since $2000$, for input values of the town’s population $p$?

(A) $f(p) = 30,000 \cdot (1.023)^p$

(B) $g(p) = \log_{1.023} \left( \frac{p}{30,000} \right)$

(D) $k(p) = 30,000 \cdot \log_{1.023} p$

▶️ Answer/Explanation

Detailed solution

The initial population at $t = 0$ is $P_0 = 30,000$.
The growth rate is $2.3\%$, so the growth factor is $1 + 0.023 = 1.023$.
The population $p$ after $t$ years is given by $p = 30,000 \cdot (1.023)^t$.
To find years $t$ as a function of population $p$, we solve for $t$:
Divide both sides by $30,000$ to get $\frac{p}{30,000} = (1.023)^t$.
Convert the exponential equation to logarithmic form: $t = \log_{1.023} \left( \frac{p}{30,000} \right)$.
Therefore, the correct function is $g(p) = \log_{1.023} \left( \frac{p}{30,000} \right)$.
The correct option is (B).

Question

The graph of $ h $ is a transformation of the function $ g $, where $ h(x) = 2g(x) + 3 $. If the point $ (3, -2) $ is on the graph of $ g $, which of the following points is on the graph of $ h^{-1} $, the inverse of $ h $?

(A) $ (-1, 3) $

(B) $ (1, 6) $

(D) $ (3, -1) $

▶️ Answer/Explanation

Detailed solution

The correct option is (A).

• The problem states that the point $ (3, -2) $ is on the graph of $ g $, which means $ g(3) = -2 $.
• We substitute $ x = 3 $ into the given equation $ h(x) = 2g(x) + 3 $ to find the corresponding point on $ h $.
• This gives $ h(3) = 2(-2) + 3 = -4 + 3 = -1 $.
• Thus, the point $ (3, -1) $ lies on the graph of the function $ h $.
• For an inverse function $ h^{-1} $, the input and output values are swapped; so if $ (a, b) $ is on $ h $, then $ (b, a) $ is on $ h^{-1} $.
• Swapping the coordinates of $ (3, -1) $ gives the point $ (-1, 3) $.
• Therefore, the point $ (-1, 3) $ is on the graph of $ h^{-1} $.

Question

The graph of the function $f$ consists of two linear segments as shown above. Which of the following is the graph of $f^{-1}$, the inverse of $f$?

(A)

(B)

(C)

(D)

▶️ Answer/Explanation

Detailed solution

The graph of $f$ contains the key endpoints and vertices: $(-1, -4)$, $(1, 0)$, and $(4, 2)$.
For an inverse function $f^{-1}$, the coordinates $(x, y)$ of $f$ are swapped to $(y, x)$.
The corresponding points for $f^{-1}$ must be $(-4, -1)$, $(0, 1)$, and $(2, 4)$.
Graph (C) is the only option that contains these specific inverted coordinates.
Graphically, $f^{-1}$ is the reflection of $f$ across the line $y = x$.
Option (C) correctly reflects the two linear segments across the identity line $y = x$.

Question (Calc Allowed)

The graph of a piecewise-linear function $f$ is shown in the figure. Let $g$ be the inverse function of $f$. What is the value of $g(7)$?

(A) $-1$
(B) $0$
(C) $1$
(D) $2$

▶️ Answer/Explanation

Detailed solution

By definition of an inverse function, if $g$ is the inverse of $f$, then $g(y) = x$ is equivalent to $f(x) = y$.
To find the value of $g(7)$, we must locate the point on the graph of $f$ where the $y$-coordinate is $7$.
Observing the provided graph, when $y = 7$, the corresponding $x$-coordinate on the line segment is $1$.
This means that $f(1) = 7$.
Therefore, $g(7) = 1$.
The correct option is (C).

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