Home / Precalculus Unit 2A MCQs Practice Questions

Question 1

The function $g$ is given by $g(x) = 25 \cdot 49^x$. Which of the following is an equivalent form for $g(x)$?
(A) $g(x) = 5 \cdot 7^{(2x)}$
(B) $g(x) = 5 \cdot 7^{(x/2)}$
(C) $g(x) = 25 \cdot 7^{(2x)}$
(D) $g(x) = 25 \cdot 7^{(x/2)}$
▶️ Answer/Explanation
Detailed solution
The correct option is (C).

The original function is given as $g(x) = 25 \cdot 49^x$.
Observe that the base $49$ can be rewritten as a power of $7$, specifically $49 = 7^2$.
Substitute this into the function: $g(x) = 25 \cdot (7^2)^x$.
Apply the power of a power rule, $(a^m)^n = a^{m \cdot n}$.
This simplifies the expression to $g(x) = 25 \cdot 7^{(2x)}$.
Comparing this result to the given options, it matches choice (C).
Therefore, the equivalent form is $g(x) = 25 \cdot 7^{(2x)}$.

Question 2

The exponential function $f$ is defined by $f(x) = ab^{x}$, where $a$ and $b$ are positive constants. The table gives values of $f(x)$ at selected values of $x$. Which of the following statements is true?
(A) $f$ demonstrates exponential decay because $a > 0$ and $0 < b < 1$.
(B) $f$ demonstrates exponential decay because $a > 0$ and $b > 1$.
(C) $f$ demonstrates exponential growth because $a > 0$ and $0 < b < 1$.
(D) $f$ demonstrates exponential growth because $a > 0$ and $b > 1$.
▶️ Answer/Explanation
Detailed solution

The correct option is (D).
Identify the $y$-intercept from the table: at $x = 0$, $f(0) = \frac{2}{3}$, so $a = \frac{2}{3}$, which is $> 0$.
Determine the common ratio $b$ by calculating $\frac{f(x+1)}{f(x)}$, for example: $\frac{f(1)}{f(0)} = \frac{2}{2/3} = 3$.
Since $b = 3$, it follows that $b > 1$.
An exponential function $f(x) = ab^{x}$ with $a > 0$ and $b > 1$ represents exponential growth.
As $x$ increases, the values of $f(x)$ in the table ($\frac{2}{9}, \frac{2}{3}, 2, 6$) are strictly increasing.
Therefore, $f$ demonstrates exponential growth because $a > 0$ and $b > 1$.

Question 3

The function $g$ is given by $g(x) = -2 \left( \frac{1}{5} \right)^x$. Which of the following limit statements is true about the graph of $g(x)$?
(A) $\lim_{x \to -\infty} g(x) = 0$ and $\lim_{x \to \infty} g(x) = \infty$
(B) $\lim_{x \to -\infty} g(x) = 0$ and $\lim_{x \to \infty} g(x) = -\infty$
(C) $\lim_{x \to -\infty} g(x) = \infty$ and $\lim_{x \to \infty} g(x) = 0$
(D) $\lim_{x \to -\infty} g(x) = -\infty$ and $\lim_{x \to \infty} g(x) = 0$
▶️ Answer/Explanation
Detailed solution

The correct option is (D).
As $x \to \infty$, the term $\left( \frac{1}{5} \right)^x$ approaches $0$, so $g(x) = -2(0) = 0$.
As $x \to -\infty$, the term $\left( \frac{1}{5} \right)^x$ is equivalent to $5^{-x}$, which becomes $5^{\infty} = \infty$.
Multiplying this positive infinity by the constant $-2$ results in $-\infty$.
Therefore, $\lim_{x \to -\infty} g(x) = -\infty$ and $\lim_{x \to \infty} g(x) = 0$.
This matches the behavior of a reflected exponential decay function.
The horizontal asymptote is the line $y = 0$.

Question 4

Values of the terms of an arithmetic sequence $a_n$ are graphed in the figure. Which of the following is an expression for the $n$th term of the arithmetic sequence?
(A) $a_n = -8 + 3(n – 1)$
(B) $a_n = -5 + 3(n – 1)$
(C) $a_n = -5 – 3(n – 1)$
(D) $a_n = -8 – 3(n – 1)$
▶️ Answer/Explanation
Detailed solution

Identify the first term $a_1$ from the graph at $n = 1$, which is $-5$.
Find the common difference $d$ by calculating the change between terms, such as $a_2 – a_1 = -2 – (-5) = 3$.
Recall the general formula for an arithmetic sequence: $a_n = a_1 + d(n – 1)$.
Substitute the identified values $a_1 = -5$ and $d = 3$ into the formula.
The resulting expression is $a_n = -5 + 3(n – 1)$.
Compare this result to the given options to find the match.
The correct option is (B).

Question 5

The value of a new boat is modeled by the function $B$. The value of the boat is expected to decrease by $11\%$ each year. At time $t = 0$ years, the value of the boat was $\$32,000$. Which of the following expressions gives the value of the boat after $m$ months? (Note: There are $12$ months in a year).
(A) $32,000(0.11)^{12m}$
(B) $32,000(0.11)^{m/12}$
(C) $32,000(0.89)^{12m}$
(D) $32,000(0.89)^{m/12}$
▶️ Answer/Explanation
Detailed solution

The initial value of the boat at $t = 0$ is $P = 32,000$.
A decrease of $11\%$ means the remaining value factor is $1 – 0.11 = 0.89$.
The annual decay formula is $V = 32,000(0.89)^t$, where $t$ is in years.
To convert years to months, we use the relation $t = \frac{m}{12}$.
Substituting this into the formula gives $V = 32,000(0.89)^{m/12}$.
Therefore, the correct expression is given in option (D).

Question 6

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 7

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) \(\frac{4x^2}{(x + 1)^2} – 4\)
(C) \(\frac{2x^2 – 4}{x + 1}\)
(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).

Question 8

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 9

Let $f$ be an exponential function of the form $f(x) = ab^{x}$, where $a$ and $b$ are positive constants. If the function $g$ is the result of a vertical dilation by a factor of $2$ to the graph of $f$, then which of the following equations could represent the function $g$?
(A) $g(x) = 2f(x)$
(B) $g(x) = f(2x)$
(C) $g(x) = \frac{1}{2}f(x)$
(D) $g(x) = f\left(\frac{x}{2}\right)$
▶️ Answer/Explanation
Detailed solution

A vertical dilation of a function $f(x)$ by a factor $k$ is represented by $g(x) = k \cdot f(x)$.
In this problem, the vertical dilation factor is given as $2$.
Applying this transformation, we multiply the entire output of the function by $2$.
This results in the equation $g(x) = 2f(x)$.
Option (B) represents a horizontal compression, not a vertical dilation.
Options (C) and (D) represent a vertical shrink and horizontal stretch, respectively.
Therefore, the correct equation for function $g$ is $g(x) = 2f(x)$.
The correct answer is (A).

Question 10

After returning home from spending spring break at the beach, Mr. Smith notices a large number of spring onions in his yard and decides to apply weed killer to his yard. The decreasing function $D$ gives the number of spring onions in his yard. The table below gives values of $D(t)$ for selected values of $t$, in weeks, since Mr. Smith applied weed killer to his yard. If a model is constructed to represent these data, which of the following best applies to this situation?
(A) $D(t) = 64 \left( \frac{1}{4} \right)^t$
(B) $D(t) = 64 \left( \frac{3}{4} \right)^t$
(C) $D(t) = 64 – \frac{1}{4}t$
(D) $D(t) = 64 – \frac{3}{4}t$
▶️ Answer/Explanation
Detailed solution

The initial value at $t = 0$ is $D(0) = 64$, which matches the coefficient in all options.
To find the rate of change, calculate the ratio between successive terms: $\frac{48}{64} = \frac{3}{4}$.
Verify the ratio for the next interval: $\frac{36}{48} = \frac{3}{4}$ and $\frac{27}{36} = \frac{3}{4}$.
Since the common ratio $r = \frac{3}{4}$ is constant, the function is exponential.
The general form for an exponential model is $D(t) = a(r)^t$.
Substituting $a = 64$ and $r = \frac{3}{4}$ gives $D(t) = 64 \left( \frac{3}{4} \right)^t$.
Therefore, the correct option is (B).

Question 11

Over the years $1950 – 2020$, the total world population has been increasing. An exponential regression was used to develop a model for the population. The figure shows a graph of the residuals of the exponential regression. Which of the following statements about the exponential regression is true?
(A) The residual plot has no apparent pattern, so the exponential model was appropriate.
(B) The residual plot has no apparent pattern, so the exponential model was not appropriate.
(C) The residual plot displays a pattern, so the exponential was appropriate.
(D) The residual plot displays a pattern, so the exponential model was not appropriate.
▶️ Answer/Explanation
Detailed solution

The correct option is (D).
The residual plot clearly shows a distinct $U$-shaped curve rather than a random scatter.
A visible pattern in a residual plot indicates that the chosen model (exponential) does not capture the underlying trend.
In statistics, a model is considered “appropriate” only if its residuals are randomly distributed around the horizontal axis.
Since a clear non-linear pattern exists, the exponential regression is an inappropriate fit for this specific data set.
Therefore, the presence of a pattern confirms the model’s inadequacy.

Question 12

The increasing function $C$ gives the number of followers, in hundreds, of a new comedian on social media. The table gives values of $C(t)$ at selected values of $t$, in months since the comedian created a social media account. If a model is constructed to represent these data, which of the following statements best applies to this situation?
(A) $C(t) = 8 \left( \frac{5}{2} \right)^t + 5$
(B) $C(t) = 8 \left( \frac{5}{2} \right)^t + 1$
(C) $C(t) = 8 \left( \frac{3}{2} \right)^t + 1$
(D) $C(t) = 8 \left( \frac{3}{2} \right)^t + 5$
▶️ Answer/Explanation
Detailed solution

At $t = 0$, the table shows $C(0) = 13$.
Substituting $t = 0$ into Option (A): $C(0) = 8(1) + 5 = 13$, which matches.
Substituting $t = 0$ into Option (B) or (C): $C(0) = 8(1) + 1 = 9$, which is incorrect.
Checking $t = 1$ for Option (A): $C(1) = 8 \left( \frac{5}{2} \right) + 5 = 20 + 5 = 25$, which matches the table.
Checking $t = 2$ for Option (A): $C(2) = 8 \left( \frac{25}{4} \right) + 5 = 50 + 5 = 55$, which matches the table.
Therefore, Option (A) correctly models the provided data points.
The base $\frac{5}{2}$ represents the growth factor of the exponential component.

Question 13 (Calc Allowed)

The graph of a geometric sequence, $g_n$, is shown above. What is the value of $g_6$ ?
(A) $0.625$
(B) $0.3125$
(C) $0.15625$
(D) $0.078125$
▶️ Answer/Explanation
Detailed solution

Identify two consecutive terms from the graph: $g_2 = 5$ and $g_3 = \frac{5}{2}$.
Calculate the common ratio $r$ using $r = \frac{g_3}{g_2} = \frac{5/2}{5} = \frac{1}{2}$ or $0.5$.
The general formula for the $n^{th}$ term is $g_n = g_2 \cdot r^{n-2}$.
Substitute the values to find $g_6$: $g_6 = 5 \cdot (0.5)^{6-2}$.
Calculate the power: $g_6 = 5 \cdot (0.5)^4 = 5 \cdot 0.0625$.
Final calculation: $g_6 = 0.3125$.
The correct option is (B).

Question 14 (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).

Question 15 (Calc Allowed)

An arithmetic sequence has the form $a_n = a_k + d(n – k)$. The graph of an arithmetic sequence is shown above. Which of the following is the corresponding linear function?
(A) $f(x) = 5 + 2(x – 3)$
(B) $f(x) = 2 + 5(x – 3)$
(C) $f(x) = 5 – 2(x – 3)$
(D) $f(x) = 2 – 5(x – 3)$
▶️ Answer/Explanation
Detailed solution

Identify two points from the graph: $(1, 1)$ and $(2, 3)$.
Calculate the common difference (slope) using $d = \frac{a_2 – a_1}{2 – 1} = \frac{3 – 1}{1} = 2$.
Locate the value of the sequence at $k = 3$, which is the point $(3, 5)$, so $a_3 = 5$.
Substitute $a_k = 5$, $d = 2$, and $k = 3$ into the formula $f(x) = a_k + d(x – k)$.
The resulting linear function is $f(x) = 5 + 2(x – 3)$.
Comparing this to the given choices, the correct option is (A).

Question 16 (Calc Allowed)

The growth of bacteria in a culture is modeled by $y = 2.3e^{1.5t}$, where $t$ is measured in days. At what time $t$ is the number of bacteria approximately $210$?
(A) $t = 3.009 \text{ days}$
(B) $t = 3.063 \text{ days}$
(C) $t = 5.612 \text{ days}$
(D) $t = 5.758 \text{ days}$
▶️ Answer/Explanation
Detailed solution

Set the bacteria count $y = 210$ in the given model: $210 = 2.3e^{1.5t}$.
Divide both sides by $2.3$ to isolate the exponential term: $\frac{210}{2.3} = e^{1.5t}$.
Calculate the ratio: $e^{1.5t} \approx 91.304$.
Take the natural logarithm ($\ln$) of both sides: $\ln(91.304) = 1.5t$.
Solve for $t$ by dividing by $1.5$: $t = \frac{\ln(91.304)}{1.5}$.
Using a calculator: $t \approx \frac{4.5142}{1.5} \approx 3.0094$.
The time $t$ is approximately $3.009 \text{ days}$.
The correct option is (A).

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