AP Precalculus -2.7 Composition of Functions- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -2.7 Composition of Functions- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -2.7 Composition of Functions- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question

The functions $ f $ and $ g $ are given by $ f(x) = x^2 + 1 $ and $ g(x) = 4x – 1 $. Which of the following is an expression for $ f(g(x)) $?

(A) $ x^2 + 4x $
(B) $ 4x^2 + 3 $
(C) $ 16x^2 – 8x + 2 $
(D) $ 4x^3 – x^2 + 4x – 1 $

▶️ Answer/Explanation

Detailed solution

Given $ f(x) = x^2 + 1 $ and $ g(x) = 4x – 1 $.
Then $ f(g(x)) = f(4x – 1) = (4x – 1)^2 + 1 $.
Expand: $ (4x – 1)^2 = 16x^2 – 8x + 1 $.
Thus, $ f(g(x)) = 16x^2 – 8x + 1 + 1 = 16x^2 – 8x + 2 $.
✅ Answer: (C)

Question

Two functions $ f $ and $ g $ are composed such that $ f(g(x)) = 4\sqrt{x-3} + 2 $. Which of the following could be the expressions for $ f(x) $ and $ g(x) $?

(A) $ f(x) = \sqrt{x} $ and $ g(x) = 4x – 3 $
(B) $ f(x) = 4x – 3 $ and $ g(x) = \sqrt{x} $
(C) $ f(x) = \sqrt{x-3} $ and $ g(x) = 4x + 2 $
(D) $ f(x) = 4x + 2 $ and $ g(x) = \sqrt{x-3} $

▶️ Answer/Explanation

Detailed solution

We have $ f(g(x)) = 4\sqrt{x-3} + 2 $.
Check option (D): $ f(x) = 4x + 2 $ and $ g(x) = \sqrt{x-3} $.
Then $ f(g(x)) = f(\sqrt{x-3}) = 4(\sqrt{x-3}) + 2 $, which matches the given composition exactly.
None of the other options produce $ 4\sqrt{x-3} + 2 $ when composed in the given order.
✅ Answer: (D)

Question

The functions $ f $ and $ g $ are given by $ f(x) = \frac{1}{x} $ and $ g(x) = \sqrt{x} $. What is the domain of the function $ h $ given by $ h(x) = f(g(x)) $?

(A) All real numbers
(B) All real numbers not equal to 0
(C) All real numbers greater than 0
(D) All real numbers greater than or equal to 0

▶️ Answer/Explanation

Detailed solution

First find $ h(x) $:
$ g(x) = \sqrt{x} $ has domain $ x \ge 0 $.
$ h(x) = f(g(x)) = f(\sqrt{x}) = \frac{1}{\sqrt{x}} $.
This requires $ \sqrt{x} > 0 $ (since division by zero is undefined).
Thus $ x > 0 $.
Domain: all real numbers greater than 0.
✅ Answer: (C)

Question

The piecewise-linear function $ f $, defined on $ -4 \le x \le 4 $, is shown in the graph. The function $ g $ is given by $ g(x) = x + 3 $. Which of the following is the graph of $ y = f(g(x)) $?

▶️ Answer/Explanation

Detailed solution

Given $ g(x) = x + 3 $, we have $ f(g(x)) = f(x + 3) $.
This composition corresponds to shifting the graph of $ f $ horizontally to the left by 3 units.
The original domain of $ f $ is $ -4 \le x \le 4 $.
For $ f(x + 3) $, we require $ -4 \le x + 3 \le 4 $, which simplifies to $ -7 \le x \le 1 $.
Thus, the graph of $ y = f(g(x)) $ is the graph of $ f $ shifted left 3 units, with the domain shifted accordingly.
✅ Answer: (A)

Question

The function $ p $ (not shown) is a polynomial function of degree 3. The graphs of four functions $ f, g, h, $ and $ k $ are given. The output values of $ p $ are the same as the output values of the composition function when $ p $ is composed with one of these functions as the input function. For which of the functions is this statement true?

(A) $ f $
(B) $ g $
(C) $ h $
(D) $ k $

▶️ Answer/Explanation

Detailed solution

The condition means $ p(f(x)) = p(x) $ for all $ x $, so $ f $ must be the identity function.
Only the graph of $ f $ is a straight line through the origin with slope 1, which represents $ f(x) = x $.
Thus, composing $ p $ with $ f $ leaves $ p $ unchanged.
✅ Answer: (A)

Question

The table gives values of the function $ f $ for selected values of $ x $. The function $ g $ is given by $ g(x) = x^2 $. Which of the following gives values of $ g(f(x)) $ for $ x = -1, x = 0, $ and $ x = 1 $?

(A) \[\begin{array}{c|c|c|c} x & -1 & 0 & 1 \\ \hline g(f(x)) & -1 & 1 & -1 \\ \end{array}\]
(B) \[\begin{array}{c|c|c|c} x & -1 & 0 & 1 \\ \hline g(f(x)) & 0 & 0 & -1 \\ \end{array}\]
(C) \[\begin{array}{c|c|c|c} x & -1 & 0 & 1 \\ \hline g(f(x)) & 0 & 1 & 1 \\ \end{array}\]
(D) \[\begin{array}{c|c|c|c} x & -1 & 0 & 1 \\ \hline g(f(x)) & 1 & 1 & 0 \\ \end{array}\]

▶️ Answer/Explanation

Detailed solution

Given $ g(x) = x^2 $, we compute $ g(f(x)) = [f(x)]^2 $.
From the table:
$ f(-1) = 0 \Rightarrow g(f(-1)) = 0^2 = 0 $
$ f(0) = 1 \Rightarrow g(f(0)) = 1^2 = 1 $
$ f(1) = -1 \Rightarrow g(f(1)) = (-1)^2 = 1 $
Thus, the values are 0, 1, 1.
✅ Answer: (C)

Question

The function $ f $ is given by $ f(x) = x^2 + 1 $, and the function $ g $ is given by $ g(x) = \frac{x-3}{x} $. Which of the following is an expression for $ f(g(x)) $?

(A) $ \frac{x^3 – 3x^2 + x – 3}{x} $
(B) $ \frac{x^2 – 2}{x^2 + 1} $
(C) $ \frac{x^2 – 6x + 9}{x^2} + 1 $
(D) $ \frac{x^2 – 8}{x^2} $

▶️ Answer/Explanation

Detailed solution

$ g(x) = \frac{x-3}{x} $
$ f(g(x)) = \left( \frac{x-3}{x} \right)^2 + 1 $
$ = \frac{(x-3)^2}{x^2} + 1 $
Expand numerator: $ (x-3)^2 = x^2 – 6x + 9 $
So $ f(g(x)) = \frac{x^2 – 6x + 9}{x^2} + 1 $
✅ Answer: (C)

Question

The table gives values of the function $ g $ for selected values of $ x $. The function $ f $ is given by $ f(x) = 3^x + x^2 $. What is the value of $ f(g(3)) $?

\[ \begin{array}{c|c} x & g(x) \\ \hline -2 & 4 \\ 0 & \frac{1}{2} \\ 3 & -2 \\ 4 & 3 \\ 36 & 9 \\ \end{array} \]
(A) $-72$
(B) $\frac{37}{9}$
(C) 9
(D) 97

▶️ Answer/Explanation

Detailed solution

From table: $ g(3) = -2 $.
Then $ f(g(3)) = f(-2) = 3^{-2} + (-2)^2 $
$ = \frac{1}{9} + 4 = \frac{1}{9} + \frac{36}{9} = \frac{37}{9} $.
✅ Answer: (B)

Question

The function $f$ is given by $f(t)=e^{t}$, and the function $g$ is given by $g(t)=7 \ln t$. If the function $h$ is given by $h(t)=(f\circ g)(t)$, which of the following is an expression for $h(t)$, for $t > 0$?

(A) $7t$
(B) $t^{7}$
(C) $te^{7}$
(D) $7$

▶️ Answer/Explanation

Detailed solution

1. Perform Composition:
$h(t) = f(g(t)) = f(7\ln t) = e^{7\ln t}$.

2. Simplify Exponent:
Use log rule $a \ln b = \ln(b^a)$: $e^{\ln(t^7)}$.

3. Evaluate:
$e^{\ln(t^7)} = t^7$.

✅ Answer: (B)

Question

The function $C$ models temperature, in degrees Celsius, as a function of time $t$, in hours, for $t\ge0.$ The function $P$ models electricity usage, in kilowatts, as a function of temperature, in degrees Celsius. Let $K$ be the composition function defined by $K(t)=P(C(t))$. Which of the following statements is true about function $K$?

(A) $K$ models electricity usage as a function of time.
(B) $K$ models temperature as a function of electricity usage.
(C) $K$ models time as a function of electricity usage.
(D) $K$ models electricity usage as a function of temperature.

▶️ Answer/Explanation

Detailed solution

1. Analyze Inner Function $C(t)$:
Input: Time ($t$).
Output: Temperature.

2. Analyze Outer Function $P(\text{Temperature})$:
Input: Temperature (which is the output of $C$).
Output: Electricity usage.

3. Analyze Composition $K(t) = P(C(t))$:
The ultimate input is $t$ (Time).
The ultimate output is the result of $P$ (Electricity usage).
Therefore, $K$ models electricity usage as a function of time.

✅ Answer: (A)

Question

The function $ f $ is given by $ f(x) = x^2 + 1 $, and the function $ g $ is given by $ g(x) = \frac{x-3}{x} $. Which of the following is an expression for $ f(g(x)) $?

(A) $ \displaystyle \frac{x^3 – 3x^2 + x – 3}{x} $
(B) $ \displaystyle \frac{x^2 – 2}{x^2 + 1} $
(C) $ \displaystyle \frac{x^2 – 6x + 9}{x^2} + 1 $
(D) $ \displaystyle \frac{x^2 – 8}{x^2} $

▶️ Answer/Explanation

Detailed solution

\[ f(g(x)) = \left( g(x) \right)^2 + 1 = \left( \frac{x-3}{x} \right)^2 + 1 \]
Simplify:
\[ = \frac{(x-3)^2}{x^2} + 1 \]
\[ = \frac{x^2 – 6x + 9}{x^2} + 1 \]
This matches option (C).
✅ Answer: (C)

Question

The table gives values for the functions $ f $ and $ g $ at selected values of $ x $. Functions $ f $ and $ g $ are defined for all real numbers. Let $ h $ be the function defined by $ h(x) = f(g(x)) $. What is the value of $ h(0) $?

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

▶️ Answer/Explanation

Detailed solution

From the (implied) table in the original problem:
\[ g(0) = 1, \quad f(1) = -2 \]
Thus, \[ h(0) = f(g(0)) = f(1) = -2 \]
✅ Answer: (A)

Question

If the function $f(x) = \sqrt{x}$ and $g(x) = \frac{1}{x-2}$ were combined according to the descriptions below, then which of the following would the combined function have a domain of $[0, 2) \cup (2, \infty)$?

i. $f(x) + g(x)$
ii. $(g \circ f)(x)$
iii. $\frac{g(x)}{f(x)}$

a. i only
b. ii only
c. i and iii only
d. i, ii, and iii

▶️ Answer/Explanation

Detailed solution

For i: $f(x) + g(x)$ requires $x \ge 0$ and $x \neq 2$, resulting in $[0, 2) \cup (2, \infty)$.
For ii: $(g \circ f)(x) = \frac{1}{\sqrt{x}-2}$ requires $x \ge 0$ and $\sqrt{x} \neq 2$, so $x \neq 4$.
For iii: $\frac{g(x)}{f(x)} = \frac{1}{\sqrt{x}(x-2)}$ requires $x > 0$ and $x \neq 2$, making the domain $(0, 2) \cup (2, \infty)$.
Only the first combination matches the required domain precisely.
The correct option is a.

Question

If $f(x) = x^4 + 3x^3 + 2x^2 + 5x + 3$ and $g(x) = -2x^8 + x^3 + 9$, then what is the end behavior of $h(x) = f(g(x))$?

a. $\lim_{x \to \infty} h(x) = \infty$ and $\lim_{x \to -\infty} h(x) = \infty$
b. $\lim_{x \to \infty} h(x) = -\infty$ and $\lim_{x \to -\infty} h(x) = -\infty$
c. $\lim_{x \to \infty} h(x) = \infty$ and $\lim_{x \to -\infty} h(x) = -\infty$
d. $\lim_{x \to \infty} h(x) = -\infty$ and $\lim_{x \to -\infty} h(x) = \infty$

▶️ Answer/Explanation

Detailed solution

The leading term of $g(x)$ is $-2x^8$, so as $x \to \pm\infty$, $g(x) \to -\infty$.
The composite function is $h(x) = f(g(x))$, where the leading term of $f(x)$ is $x^4$.
As $x \to \infty$, $h(x) \approx f(-2x^8) \approx (-2x^8)^4 = 16x^{32}$, which approaches $\infty$.
As $x \to -\infty$, $h(x) \approx f(-2x^8) \approx (-2x^8)^4 = 16x^{32}$, which approaches $\infty$.
Since both ends approach positive infinity, the correct option is a.
Final Result: $\lim_{x \to \infty} h(x) = \infty$ and $\lim_{x \to -\infty} h(x) = \infty$.

Question

The table gives values for the functions $f$ and $g$ at selected values of $x$. Functions $f$ and $g$ are defined for all real numbers.

Let $h$ be the function defined by $h(x) = g(f(x))$. What is the value of $h(9)$?

(A) $4$
(B) $7$
(C) $8$
(D) $10$

▶️ Answer/Explanation

Detailed solution

To find the value of $h(9)$, we use the composition formula $h(9) = g(f(9))$.
First, locate $x = 9$ in the table and identify the value of $f(9)$.
From the table, when $x = 9$, $f(9) = 5$.
Substitute this value into the outer function: $g(f(9)) = g(5)$.
Next, locate $x = 5$ in the table and identify the value of $g(5)$.
From the table, when $x = 5$, $g(5) = 4$.
Therefore, $h(9) = 4$, which corresponds to option (A).

Question

The table gives values for the functions $f$ and $g$ at selected values of $x$. Functions $f$ and $g$ are defined for all real numbers.

Let $k$ be the function defined by $k(x) = f(g(x))$. What is the value of $k(3)$?

(A) $3$
(B) $7$
(C) $8$
(D) $10$

▶️ Answer/Explanation

Detailed solution

The correct answer is (C).
To find $k(3)$, identify the composite function $k(3) = f(g(3))$.
First, locate $x = 3$ in the table to find the value of $g(3)$.
From the table, when $x = 3$, $g(3) = 5$.
Substitute this value into the outer function: $f(g(3)) = f(5)$.
Next, locate $x = 5$ in the table to find the value of $f(5)$.
From the table, when $x = 5$, $f(5) = 8$.
Therefore, $k(3) = 8$.

Question

The graphs of $f$ and $g$ are shown for $0 \leq x \leq 9$, each consisting of three line segments.

Let $h$ be the function defined by $h(x) = g(f(x))$. What is the value of $h(8)$?

(A) $-2$

(B) $-1$

▶️ Answer/Explanation

Detailed solution

To find $h(8)$, use the definition of composite functions: $h(8) = g(f(8))$.
First, locate $x = 8$ on the horizontal axis of the Graph of $f$.
Move vertically to the line segment to find the $y$-value: $f(8) = 1$.
Substitute this result into the outer function to get $g(1)$.
Now, locate $x = 1$ on the horizontal axis of the Graph of $g$.
Move vertically to the line segment to find the corresponding $y$-value: $g(1) = -2$.
Therefore, $h(8) = g(1) = -2$, which corresponds to choice (A).

Question

The graphs of $f$ and $g$ are shown for $0 \leq x \leq 9$, each consisting of three line segments.

Let $k$ be the function defined by $k(x) = f(g(x))$. What is the value of $k(9)$?

(A) $-3$

(B) $-2$

(D) $4$

▶️ Answer/Explanation

Detailed solution

To find $k(9)$, we use the definition of a composite function: $k(9) = f(g(9))$.
First, identify the value of $g(9)$ by looking at the Graph of $g$ at $x = 9$.
From the graph, the point $(9, 3)$ exists, so $g(9) = 3$.
Next, substitute this value into $f$: $k(9) = f(3)$.
Identify the value of $f(3)$ by looking at the Graph of $f$ at $x = 3$.
From the graph, the point $(3, -2)$ exists, so $f(3) = -2$.
Therefore, $k(9) = -2$, which corresponds to option (B).

Question

The table gives values of the function $f$ for selected values of $x$. The function $g$ is given by $g(x) = 9^x – 8x$.

What is the value of $g(f(1))$?

(A) $-1$

(B) $\frac{1}{2}$

(D) $65$

▶️ Answer/Explanation

Detailed solution

To find $g(f(1))$, first identify the value of $f(1)$ from the provided table.
From the table, when $x = 1$, the corresponding value is $f(1) = \frac{1}{2}$.
Next, substitute this value into the function $g(x) = 9^x – 8x$.
This gives the expression $g\left(\frac{1}{2}\right) = 9^{1/2} – 8\left(\frac{1}{2}\right)$.
Calculate the square root: $9^{1/2} = \sqrt{9} = 3$.
Calculate the product: $8 \cdot \frac{1}{2} = 4$.
Subtract the values: $3 – 4 = -1$.
Therefore, the correct option is (A).

Question

The function $f$ is given by $f(x) = 4^{x^2}$, and the function $g$ is given by $g(x) = 3x$. Which of the following is an expression for $f(g(x))$?

(A) $4^{3x^2}$

(B) $4^{6x^2}$

(D) $3 \cdot 4^{x^2}$

▶️ Answer/Explanation

Detailed solution

To find the composite function $f(g(x))$, substitute the expression for $g(x)$ into $f(x)$.
Given $g(x) = 3x$, we replace the $x$ in $f(x) = 4^{x^2}$ with $(3x)$.
This gives the expression $f(g(x)) = 4^{(3x)^2}$.
Apply the power of a product rule: $(3x)^2 = 3^2 \cdot x^2$.
Simplify the exponent: $3^2 = 9$, resulting in $9x^2$.
The final expression is $4^{9x^2}$.
Therefore, the correct choice is (C).

Question

The function $f$ is given by $f(x) = 2x – 1$, and the function $g$ is given by $g(x) = x^2 + 3x$. Which of the following is an expression for $g(f(x))$?

(A) $2x^2 + 6x – 1$

(B) $4x^2 – x – 1$

(D) $2x^3 + 5x^2 – 3x$

▶️ Answer/Explanation

Detailed solution

To find the composite function $g(f(x))$, substitute the expression for $f(x)$ into $g(x)$.
Substitute $f(x) = 2x – 1$ into $g(x) = x^2 + 3x$:
$g(f(x)) = (2x – 1)^2 + 3(2x – 1)$
Expand the squared term: $(2x – 1)^2 = 4x^2 – 4x + 1$
Distribute the 3: $3(2x – 1) = 6x – 3$
Combine the parts: $4x^2 – 4x + 1 + 6x – 3$
Simplify by combining like terms: $4x^2 + 2x – 2$
Therefore, the correct choice is (C).

Question

The function $ f $ is given by $ f(t) = e^t $, and the function $ g $ is given by $ g(t) = 7 \ln t $. If the function $ h $ is given by $ h(t) = (f \circ g)(t) $, which of the following expression for $ h(t) $, for $ t > 0 $?

(A) $ 7t $

(B) $ t^7 $

(D) $ 7^t $

▶️ Answer/Explanation

Detailed solution

The correct answer is (B).
We need to evaluate the composite function $ h(t) = f(g(t)) $.
Substitute $ g(t) = 7 \ln t $ into the function $ f $:
$ h(t) = f(7 \ln t) = e^{7 \ln t} $
Apply the power rule for logarithms, $ a \ln b = \ln(b^a) $, to the exponent:
$ h(t) = e^{\ln(t^7)} $
Use the inverse property of exponential and logarithmic functions, $ e^{\ln x} = x $:
$ h(t) = t^7 $

Question

The function $ f $ is given by $ f(x) = e^{2x} $, and the function $ g $ is given by $ g(x) = \ln(3x) $. For $ x > 0 $, which of the following is an expression for $ f(g(x)) $?

(A) $ 9x^2 $

(B) $ 2x + \ln 3 $

(D) $ 6x $

▶️ Answer/Explanation

Detailed solution

To find the composite function $ f(g(x)) $, substitute $ g(x) $ into $ f(x) $.

Given $ f(x) = e^{2x} $ and $ g(x) = \ln(3x) $, substitute $ \ln(3x) $ for $ x $ in $ f $:

$$ f(g(x)) = e^{2 \ln(3x)} $$

Use the power property of logarithms, $ a \ln b = \ln(b^a) $, to rewrite the exponent:

$$ 2 \ln(3x) = \ln((3x)^2) = \ln(9x^2) $$

Now, the expression becomes $ f(g(x)) = e^{\ln(9x^2)} $.

Apply the inverse property of natural logarithms and exponentials, $ e^{\ln u} = u $:

$$ f(g(x)) = 9x^2 $$

Therefore, the correct expression is (A).

Question

The function $f$ is given by $f(t) = e^{t}$, and the function $g$ is given by $g(t) = 7 \ln t$. If the function $h$ is given by $h(t) = (f \circ g)(t)$, which of the following is an expression for $h(t)$, for $t > 0$?

(A) $7t$

(B) $t^{7}$

(D) $7^{t}$

▶️ Answer/Explanation

Detailed solution

The composition is defined as $h(t) = f(g(t))$.
Substitute $g(t) = 7 \ln t$ into $f(t)$, resulting in $h(t) = e^{7 \ln t}$.
Apply the power rule for logarithms: $7 \ln t = \ln(t^{7})$.
The expression becomes $h(t) = e^{\ln(t^{7})}$.
Using the identity $e^{\ln x} = x$, the expression simplifies to $t^{7}$.
Therefore, the correct expression for $h(t)$ is $t^{7}$.
The correct option is (B).

Question

Both nonzero functions $f$ and $g$ are invertible. The input values of $f$ are times, in hours, and the output values of $f$ are rates, in miles per hour. The input values of $g$ are rates, in miles per hour, and the output values of $g$ are costs, in dollars. For which of the following are the input values costs, in dollars, and the output values times, in hours?

(A) $y = f(g(x))$

(B) $y = g^{-1}(f^{-1}(x))$

(D) $y = g(f(x))$

▶️ Answer/Explanation

Detailed solution

For function $f$, the mapping is $\text{hours} \rightarrow \text{miles per hour}$.
For function $g$, the mapping is $\text{miles per hour} \rightarrow \text{dollars}$.
The inverse $g^{-1}$ reverses its mapping to $\text{dollars} \rightarrow \text{miles per hour}$.
The inverse $f^{-1}$ reverses its mapping to $\text{miles per hour} \rightarrow \text{hours}$.
To start with dollars, $g^{-1}(x)$ must be the inner function.
The output of $g^{-1}$ matches the required input for $f^{-1}$.
Thus, $y = f^{-1}(g^{-1}(x))$ maps dollars to hours.
The correct option is (C).

Question

Consider the constant function $f$ given by $f(x) = -1$ and the function $g$ given by $g(x) = \log_{3} x$. Let $h$ be the function given by $h(x) = g(x) – f(x)$. In the $xy$-plane, what is the $x$-intercept of the graph of $h$?

(A) $(-3, 0)$
(B) $(\frac{1}{3}, 0)$
(C) $(3, 0)$
(D) The graph of $h$ does not have an $x$-intercept.

▶️ Answer/Explanation

Detailed solution

First, define the function $h(x) = g(x) – f(x)$.
Substitute the given functions: $h(x) = \log_{3} x – (-1)$, which simplifies to $h(x) = \log_{3} x + 1$.
To find the $x$-intercept, set $h(x) = 0$.
This gives the equation $0 = \log_{3} x + 1$.
Subtract $1$ from both sides to get $\log_{3} x = -1$.
Rewrite the logarithmic equation in exponential form: $x = 3^{-1}$.
Solving for $x$ yields $x = \frac{1}{3}$.
The $x$-intercept is the point $(\frac{1}{3}, 0)$, which corresponds to option (B).

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) $2$
(C) $6$
(D) $12$

▶️ Answer/Explanation

Detailed solution

First, locate $x = 3$ in the table to find the value of $k(3)$.
From the table, when $x = 3$, the value of $k(3) = 4$.
The expression now becomes $h^{-1}(4)$.
To find $h^{-1}(4)$, look for the value of $x$ such that $h(x) = 4$.
From the table, $h(x) = 4$ when $x = 1$.
Therefore, $h^{-1}(4) = 1$.
The final value is $1$, which corresponds to option (A).

Question

The table gives values for the functions $f$ and $g$ at selected values of $x$. Functions $f$ and $g$ are defined for all real numbers. Let $h$ be the function defined by $h(x) = g(f(x))$. What is the value of $h(4)$?

(A) $-3$
(B) $1$
(C) $2$
(D) $7$

▶️ Answer/Explanation

Detailed solution

To find $h(4)$, we use the definition $h(4) = g(f(4))$.
First, locate $x = 4$ in the table to find the value of $f(4)$.
From the table, when $x = 4$, $f(4) = 3$.
Now, substitute this value into the function $g$, so we need to find $g(3)$.
Locate $x = 3$ in the table to find the value of $g(3)$.
From the table, when $x = 3$, $g(3) = 7$.
Therefore, $h(4) = 7$.
The correct option is (D).

Question

The function $f$ is given by $f(x) = x^2 – 4$, and the function $g$ is given by $g(x) = \frac{2x}{x+1}$. Which of the following is an expression for $g(f(x))$?

(A) $\frac{2x^3 – 8x}{x + 1}$

(B) $\frac{4x^2}{(x + 1)^2} – 4$

(D) $\frac{2(x^2 – 4)}{x^2 – 3}$

▶️ Answer/Explanation

Detailed solution

To find $g(f(x))$, substitute the expression for $f(x)$ into the function $g(x)$.
Given $f(x) = x^2 – 4$, replace every $x$ in $g(x) = \frac{2x}{x+1}$ with $(x^2 – 4)$.
The numerator becomes: $2(x^2 – 4)$.
The denominator becomes: $(x^2 – 4) + 1$.
Simplify the denominator: $-4 + 1 = -3$, resulting in $x^2 – 3$.
The final composite expression is $\frac{2(x^2 – 4)}{x^2 – 3}$.
Therefore, the correct choice is (D).

AP Precalculus -2.7 Composition of Functions- MCQ Exam Style Questions - Effective Fall 2023