AP Precalculus -2.4 Exponential Function Manipulation- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -2.4 Exponential Function Manipulation- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -2.4 Exponential Function Manipulation- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question

Iodine-131 has a half-life of 8 days. In a particular sample, the amount of iodine-131 remaining after $ d $ days can be modeled by the function $ h $ given by $ h(d) = A_0(0.5)^{(d/8)} $, where $ A_0 $ is the amount of iodine-131 in the sample at time $ d = 0 $. Which of the following functions $ k $ models the amount of iodine-131 remaining after $ t $ hours, where $ A_0 $ is the amount of iodine-131 in the sample at time $ t = 0 $? (There are 24 hours in a day, so $ t = 24d $.)

(A) $ k(t) = A_0(0.5)^{(t/24)} $
(B) $ k(t) = A_0\left(0.5^{(1/24)}\right)^{(8t)} $
(C) $ k(t) = A_0\left(0.5^{(24)}\right)^{(t/8)} $
(D) $ k(t) = A_0\left(0.5^{(1/192)}\right)^t $

▶️ Answer/Explanation

Detailed solution

Given $ h(d) = A_0(0.5)^{d/8} $ with $ t = 24d $, so $ d = \frac{t}{24} $.
Substitute into $ h $:
$ k(t) = A_0(0.5)^{\frac{t/24}{8}} = A_0(0.5)^{t/(192)} $
Rewrite exponent: $ 0.5^{t/192} = \left(0.5^{1/192}\right)^t $.
Thus $ k(t) = A_0\left(0.5^{1/192}\right)^t $.
✅ Answer: (D)

Question

The value, in millions of dollars, of transactions processed by an online payment platform is modeled by the function $ M $. The value is expected to increase by 6.1% each quarter of a year. At time $ t = 0 $ years, 54 million dollars of transactions were processed. If $ t $ is measured in years, which of the following is an expression for $ M(t) $?
(Note: A quarter is one fourth of a year.)

(A) $ 54(0.061)^{(t/4)} $
(B) $ 54(0.061)^{(4t)} $
(C) $ 54(1.061)^{(t/4)} $
(D) $ 54(1.061)^{(4t)} $

▶️ Answer/Explanation

Detailed solution

Growth per quarter: multiply by $ 1 + 0.061 = 1.061 $.
Each year has 4 quarters, so in $ t $ years there are $ 4t $ quarters.
Starting value $ A_0 = 54 $.
Thus $ M(t) = 54 \cdot (1.061)^{4t} $.
✅ Answer: (D)

Question

The function $ f $ is given by $ f(x) = 2^x $, and the function $ g $ is given by $ g(x) = \frac{f(x)}{8} $. For which of the following transformations is the graph of $ g $ the image of the graph of $ f $?

(A) A horizontal translation to the left 3 units
(B) A horizontal translation to the right 3 units
(C) A vertical translation up $ \frac{1}{8} $ unit
(D) A vertical translation down $ \frac{1}{8} $ unit

▶️ Answer/Explanation

Detailed solution

$ g(x) = \frac{2^x}{8} = \frac{2^x}{2^3} = 2^{x-3} $.
This is $ f(x-3) $, which corresponds to shifting the graph of $ f $ horizontally to the right by 3 units.
✅ Answer: (B)

Question

The function $ m $ is given by $ m(x) = 36^{(x/2)} $. Which of the following expressions could also define $ m(x) $?

(A) $ 6^x $
(B) $ 6 \cdot 6^x $
(C) $ 18^x $
(D) $ 18 \cdot 36^x $

▶️ Answer/Explanation

Detailed solution

$ 36^{(x/2)} = (36^{1/2})^x $.
Since $ 36^{1/2} = 6 $, we have $ 6^x $.
✅ Answer: (A)

Question

The function $ k $ is given by $ k(x) = 9^x $. Which of the following expressions also defines $ k(x) $?

(A) $ 2^{(3x)} $
(B) $ 3^{(2x)} $
(C) $ 3^{(3x)} $
(D) $ 3^{(x/2)} $

▶️ Answer/Explanation

Detailed solution

$ 9^x = (3^2)^x = 3^{2x} $.
✅ Answer: (B)

Question

The function $ f $ is given by $ f(x) = 3^x $. The function $ g $ is given by $ g(x) = (f(x))^b $, where $ b < 0 $. Which of the following describes the relationship between the graphs of $ f $ and $ g $?

(A) The graph of $ g $ is a combination of a horizontal dilation of the graph of $ f $ and a reflection over the $ x $-axis.
(B) The graph of $ g $ is a combination of a horizontal dilation of the graph of $ f $ and a reflection over the $ y $-axis.
(C) The graph of $ g $ is a combination of a vertical dilation of the graph of $ f $ and a reflection over the $ x $-axis.
(D) The graph of $ g $ is a combination of a vertical dilation of the graph of $ f $ and a reflection over the $ y $-axis.

▶️ Answer/Explanation

Detailed solution

$ g(x) = (3^x)^b = 3^{bx} $, with $ b < 0 $.
Let $ b = -c $ where $ c > 0 $, then $ g(x) = 3^{-cx} = (3^{-c})^x $.
The transformation $ x \to -cx $ represents a horizontal stretch/compression (dilation) and a reflection over the $ y $-axis (due to negative sign).
✅ Answer: (B)

Question

The functions $ f $ and $ g $ are given by $ f(x) = 2^x $ and $ g(x) = 2^x \cdot 2^a $, where $ a > 0 $. Which of the following describes the relationship between the graph of $ f $ and the graph of $ g $?

(A) The graph of $ g $ is a vertical translation of the graph of $ f $ by $ a $ units.
(B) The graph of $ g $ is a horizontal translation of the graph of $ f $ by $ a $ units.
(C) The graph of $ g $ is a vertical translation of the graph of $ f $ by $ -a $ units.
(D) The graph of $ g $ is a horizontal translation of the graph of $ f $ by $ -a $ units.

▶️ Answer/Explanation

Detailed solution

$ g(x) = 2^x \cdot 2^a = 2^{x+a} = f(x+a) $.
This corresponds to shifting the graph of $ f $ left by $ a $ units (horizontal translation by $ -a $ units).
✅ Answer: (D)

Question

In the $xy$-plane, the function $h$, given by $h(x)=3^{(x+2)}$, is a horizontal translation of $f(x)=3^{x}$. Which of the following is an equivalent form for $h(x)$ that expresses $h$ as a vertical dilation of $f$?

(A) $h(x)=3^{(x/2)}$
(B) $h(x)=9\cdot3^{x}$
(C) $h(x)=9\cdot(\frac{1}{3})^{x}$
(D) $h(x)=9+3^{x}$

▶️ Answer/Explanation

Detailed solution

1. Apply Exponent Properties:
$h(x) = 3^{x+2} = 3^x \cdot 3^2$.

2. Simplify:
$3^2 = 9$.
So, $h(x) = 9 \cdot 3^x$.
This represents a vertical dilation of $f$ by a factor of 9.

✅ Answer: (B)

Question

The function $ f $ is given by $ f(x) = 9 \cdot 25^x $. Which of the following is an equivalent form for $ f(x) $?

(A) $ f(x) = 3 \cdot 5^{(x/2)} $
(B) $ f(x) = 3 \cdot 5^{(2x)} $
(C) $ f(x) = 9 \cdot 5^{(x/2)} $
(D) $ f(x) = 9 \cdot 5^{(2x)} $

▶️ Answer/Explanation

Detailed solution

$ f(x) = 9 \cdot 25^x $.
Since $ 25 = 5^2 $, we can write $ 25^x = (5^2)^x = 5^{2x} $.
Thus $ f(x) = 9 \cdot 5^{2x} $.
This matches option (D).
✅ Answer: (D)

Question

The function $f$ is given by $f(x) = 36 \cdot 4^x$. Which of the following is an equivalent form for $f(x)$?

(A) $f(x) = 6 \cdot 2^{(x/2)}$

(B) $f(x) = 6 \cdot 2^{(2x)}$

(D) $f(x) = 36 \cdot 2^{(2x)}$

▶️ Answer/Explanation

Detailed solution

The given function is $f(x) = 36 \cdot 4^x$.
Identify the base of the exponential term, which is $4$.
Express the base $4$ as a power of $2$, so $4 = 2^2$.
Substitute $2^2$ back into the function: $f(x) = 36 \cdot (2^2)^x$.
Apply the power of a power rule, $(a^m)^n = a^{m \cdot n}$.
This simplifies the expression to $f(x) = 36 \cdot 2^{(2x)}$.
Comparing this to the options, the equivalent form is (D).

Question

The function $j$ is given by $j(x) = 3 \cdot 4^{(x+2)}$. Which of the following is an equivalent form for $j(x)$?

(A) $j(x) = 48^x$

(B) $j(x) = 48 \cdot 4^x$

(D) $j(x) = 9 \cdot 16^x$

▶️ Answer/Explanation

Detailed solution

The correct answer is (B).
Start with the given function: $j(x) = 3 \cdot 4^{(x+2)}$.
Apply the product rule for exponents: $a^{m+n} = a^m \cdot a^n$.
This allows us to rewrite the term $4^{(x+2)}$ as $4^2 \cdot 4^x$.
Substitute this back into the function: $j(x) = 3 \cdot 4^2 \cdot 4^x$.
Calculate the constant value: $4^2 = 16$.
Multiply the constants: $3 \cdot 16 = 48$.
The resulting equivalent form is $j(x) = 48 \cdot 4^x$.

Question

Which of the following functions is an equivalent form of the function $g(x) = 3 \cdot 5^{2x}$?

(A) $g(x) = 75^x$

(B) $g(x) = 3 \cdot 25^x$

(D) $g(x) = 9 \cdot 25^x$

▶️ Answer/Explanation

Detailed solution

The correct option is (B).
Start with the original function: $g(x) = 3 \cdot 5^{2x}$.
Apply the power of a power rule: $a^{bc} = (a^b)^c$.
Rewrite the exponential part as $(5^2)^x$.
Calculate the value inside the parentheses: $5^2 = 25$.
Substitute the value back to get $g(x) = 3 \cdot 25^x$.
This matches option (B).

Question

The function $h$ is given by $h(x) = 25 \cdot 16^{(x/2)}$. Which of the following is an equivalent form for $h(x)$?

(A) $h(x) = 100^{x}$

(B) $h(x) = 5 \cdot 4^{x}$

(D) $h(x) = 25 \cdot 8^{x}$

▶️ Answer/Explanation

Detailed solution

The given function is $h(x) = 25 \cdot 16^{(x/2)}$.
Using the exponent rule $(a^{m})^{n} = a^{m \cdot n}$, rewrite $16^{(x/2)}$ as $(16^{1/2})^{x}$.
Since $16^{1/2}$ is the square root of $16$, we find $16^{1/2} = 4$.
Substitute $4$ back into the expression to get $h(x) = 25 \cdot (4)^{x}$.
Comparing this to the options, it matches choice (C).
Therefore, the equivalent form is $h(x) = 25 \cdot 4^{x}$.

Question

Which of the following functions is an equivalent form of the function $p(x) = 4^{(-2x)}$?

(A) $p(x) = -(16)^x$

(B) $p(x) = (2)^x$

(D) $p(x) = \left(\frac{1}{16}\right)^x$

▶️ Answer/Explanation

Detailed solution

The given function is $p(x) = 4^{(-2x)}$.
Using the power of a power rule, $(a^m)^n = a^{mn}$, we can rewrite the expression as $p(x) = (4^{-2})^x$.
Apply the negative exponent rule $a^{-n} = \frac{1}{a^n}$ to the base.
This gives $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$.
Substituting this back into the function, we get $p(x) = \left(\frac{1}{16}\right)^x$.
Therefore, the equivalent form is option (D).

Question

In the $xy$-plane, the function $f$, given by $f(x) = 4^{(x-2)}$, is a horizontal translation of the exponential function $g$, given by $g(x) = 4^x$. Which of the following is an equivalent form for $f(x)$ that expresses $f$ as a vertical dilation of $g$?

(A) $f(x) = -2 \cdot 4^x$

(B) $f(x) = 16 \cdot 4^x$

(D) $f(x) = \frac{1}{16} + 4^x$

▶️ Answer/Explanation

Detailed solution

The given function is $f(x) = 4^{(x-2)}$.
By applying the exponent rule $a^{m-n} = \frac{a^m}{a^n}$, we can rewrite the expression.
$f(x) = 4^x \cdot 4^{-2}$
$f(x) = 4^x \cdot \frac{1}{4^2}$
$f(x) = \frac{1}{16} \cdot 4^x$
Since $g(x) = 4^x$, this form represents $f(x) = \frac{1}{16} \cdot g(x)$.
This shows $f$ as a vertical dilation of $g$ by a factor of $\frac{1}{16}$.
Therefore, the correct equivalent form is option (C).

Question

In the $xy$-plane, the function $h$, given by $h(x) = 9 \cdot 3^{x}$, is a vertical dilation of the exponential function $k$, given by $k(x) = 3^{x}$. Which of the following is an equivalent form for $h(x)$ that expresses $h$ as a horizontal translation of $k$?

(A) $h(x) = 3^{(2x)}$
(B) $h(x) = 3^{(x+2)}$
(C) $h(x) = 3^{(x-2)}$
(D) $h(x) = 9 + 3^{x}$

▶️ Answer/Explanation

Detailed solution

The given function is $h(x) = 9 \cdot 3^{x}$.
We recognize that $9$ can be written as a power of the base $3$, specifically $9 = 3^{2}$.
Substitute this into the expression: $h(x) = 3^{2} \cdot 3^{x}$.
Using the exponent rule $a^{m} \cdot a^{n} = a^{m+n}$, we combine the terms.
This results in $h(x) = 3^{(x+2)}$.
In the form $f(x-c)$, a value of $c = -2$ represents a horizontal translation 2 units to the left.
Therefore, the correct equivalent form is (B).

Question

The graph of a function $f$ is a horizontal dilation of $y = 3^x$, and $f(x)$ is equivalent to $(\sqrt{3})^x$. Which of the following could be an expression for $f(x)$?

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

(B) $3^{(2x)}$

(D) $9^{(2x)}$

▶️ Answer/Explanation

Detailed solution

The given function is $f(x) = (\sqrt{3})^x$.
Recall that the square root of a number can be written as an exponent of $1/2$, so $\sqrt{3} = 3^{1/2}$.
Substitute this into the expression: $f(x) = (3^{1/2})^x$.
Apply the power of a power rule, $(a^m)^n = a^{m \cdot n}$.
This simplifies the expression to $f(x) = 3^{(1/2) \cdot x}$.
Therefore, $f(x) = 3^{(x/2)}$.
This matches Option (A).

Question

Zebra mussels are an invasive, fingernail-sized mollusk that are infesting some of the freshwater lakes in North America. Their population increases on average at a rate of 5% per day. Let 700 be the amount of zebra mussels in a certain lake at time $ d = 0 $ days. Which of the following functions $ g $ models the amount of zebra mussels after $ t $ weeks where 700 is the amount of zebra mussels at time $ t = 0 $?

(A) $ g(t) = 700(1.05)^{\frac{t}{7}} $

(B) $ g(t) = 700(1.05)^{7t} $

(D) $ g(t) = 700(1.05^7)^{\frac{t}{7}} $

▶️ Answer/Explanation

Detailed solution

The correct option is (B).

The initial population is given as 700, and the growth rate is 5% per day, meaning the daily growth factor is $ 1 + 0.05 = 1.05 $.
The population after $ d $ days can be modeled by the function $ P(d) = 700(1.05)^d $.
The problem asks for a function $ g(t) $ in terms of weeks $ t $, not days.
Since there are 7 days in one week, we can convert weeks to days using the relationship $ d = 7t $.
Substituting $ 7t $ for $ d $ in the original equation gives $ g(t) = 700(1.05)^{7t} $.
Comparing this result to the choices provided, it matches option (B).

Question

In the $xy$-plane, the function $h$, given by $h(x) = 9 \cdot 3^x$, is a vertical dilation of the exponential function $k$, given by $k(x) = 3^x$. Which of the following is an equivalent form for $h(x)$ that expresses $h$ as a horizontal dilation of $k$?

(A) $h(x) = 3^{(2x)}$

(B) $h(x) = 3^{(x+2)}$

(D) $h(x) = 9 + 3^x$

▶️ Answer/Explanation

Detailed solution

The goal is to find an expression equivalent to $h(x) = 9 \cdot 3^x$ using properties of exponents.
First, we express the constant $9$ as a power with base $3$, such that $9 = 3^2$.
Next, we substitute this into the function equation: $h(x) = 3^2 \cdot 3^x$.
Using the product rule for exponents, $a^m \cdot a^n = a^{m+n}$, we combine the terms.
This simplification results in $h(x) = 3^{(2+x)}$ or $h(x) = 3^{(x+2)}$.
Comparing this result to the given options, it matches option (B).

Correct Option: (B)

Question

Iodine-131 has a half-life of 8 days. In a particular sample, the amount of iodine-131 remaining after (d) days can be modeled by the functions (h) given by (h(d) = A_0(0.5)^{(d/8)}), where (A_0) is the amount of iodine-131 in the sample at time (d = 0). Which of the following functions (k) models the amount of iodine-131 remaining after (t) hours, where (A_0) is the amount of iodine-131 in the sample at time (t = 0)? (There are 24 hours in a day, so (t = 24d).)

(A) (k(t) = A_0(0.5)^{(t/24)}) (B) (k(t) = A_0(0.5^{(1/24)})^{(8t)}) (C) (k(t) = A_0(0.5^{(24)})^{(t/8)}) (D) (k(t) = A_0(0.5^{(1/192)})^{(t/8)})

▶️ Answer/Explanation

Detailed solution

We are given the model (h(d) = A_0(0.5)^{d/8}) where (d) is in days. We need to convert the variable to hours (t), given that (t = 24d). First, solve for (d) in terms of (t): (d = \frac{t}{24}). Substitute this expression for (d) into the original function: (k(t) = A_0(0.5)^{\frac{t/24}{8}}). Simplify the exponent by multiplying the denominators: (\frac{t}{24 \cdot 8} = \frac{t}{192}). Therefore, the function is (k(t) = A_0(0.5)^{t/192}). Comparing this to the options, Option (D) is the correct choice as it utilizes the factor (192) derived from (24 \times 8).

Question

The function $ f $ is given by $ f(x) = 2^{(3x)} $. Which of the following statements describes characteristics of the graph of $ f $ in the $ xy $-plane?

(A) The graph of $ f $ is a vertical dilation of $ y = 2^x $, and $ f(x) $ is equivalent to $ 8^x $.

(B) The graph of $ f $ is a vertical dilation of $ y = 2^x $, and $ f(x) $ is equivalent to $ 8 \cdot 2^x $.

(D) The graph of $ f $ is a horizontal dilation of $ y = 2^x $, and $ f(x) $ is equivalent to $ 8 \cdot 2^x $.

▶️ Answer/Explanation

Detailed solution

The correct option is (C).

The function is given by $ f(x) = 2^{(3x)} $.
Comparing this to the parent function $ y = 2^x $, the input $ x $ is multiplied by $ 3 $, which corresponds to a horizontal dilation (specifically a horizontal compression).
This distinguishes the transformation from a vertical dilation, eliminating options (A) and (B).
To find the equivalent expression, we apply the power of a power rule for exponents: $ a^{(bc)} = (a^b)^c $.
Rewriting the function: $ f(x) = (2^3)^x $.
Since $ 2^3 = 8 $, the function simplifies to $ f(x) = 8^x $.
Therefore, the graph is a horizontal dilation of $ y = 2^x $ and $ f(x) $ is equivalent to $ 8^x $.

Question

The population of people in a community is decreasing over time. The population can be modeled by the exponential decay function $P$ given by $P(t) = 20,000(0.92)^t$, where $t$ is measured in years since 2020. Based on updated data, a new exponential decay function model for the population was constructed. The new model $Q$ is given by $Q(t) = 20,000((0.9)(0.92))^t$. Which of the following describes the relationship between model $Q$ and model $P$?

(A) In model $Q$ the new growth factor is reduced by $0.9$ times the growth factor in model $P$.

(B) In model $Q$ the new initial value is reduced by $0.9$ times the initial value in model $P$.

(D) In model $Q$ the new initial value is $0.9$ times as large as the initial value in model $P$.

▶️ Answer/Explanation

Detailed solution

The correct option is (C).

The standard form of an exponential function is $y = a(b)^t$, where $a$ is the initial value and $b$ is the growth (or decay) factor.
In model $P(t) = 20,000(0.92)^t$, the initial value is $20,000$ and the growth factor is $0.92$.
In model $Q(t) = 20,000((0.9)(0.92))^t$, the initial value is $20,000$ and the growth factor is $(0.9)(0.92)$.
Comparing the two models, the initial value remains $20,000$ in both, so options (B) and (D) are incorrect.
The growth factor in model $Q$ is the product of $0.9$ and the growth factor of model $P$ ($0.92$).
This means the new growth factor is $0.9$ times as large as the original growth factor.
Option (A) suggests the factor is “reduced by,” which implies subtraction, whereas the equation shows multiplication.

Question

The functions $ f $ and $ g $ are given by $ f(x) = 4^{(5x-1)} $ and $ g(x) = 8^{(x/4)} $. When solving the equation $ f(x) = g(x) $, the functions can be rewritten in equivalent forms so that the equation can be solved without the use of technology. Which of the following are equivalent definitions of $ f $ and $ g $ that aid in solving $ f(x) = g(x) $ without the use of technology?

(A) $ f(x) = 2^{(\log_2 4 \cdot (5x-1))} $ and $ g(x) = 2^{(\log_2 8 \cdot (x/4))} $

(B) $ f(x) = 2^{(\log_2 8 \cdot (5x-1))} $ and $ g(x) = 2^{(\log_2 4 \cdot (x/4))} $

(D) $ f(x) = 2 \cdot 4^{(\log_2 4 \cdot (5x-1))} $ and $ g(x) = 8^{(\log_2 8 \cdot (x/4))} $

▶️ Answer/Explanation

Detailed solution

The correct answer is (A). To solve the equation without technology, we must rewrite the functions using a common base. Both 4 and 8 are powers of 2.

1. Identify the common base: Since $ 4 = 2^2 $ and $ 8 = 2^3 $, we use base 2.
2. Rewrite base 4 using logarithms: $ 4 = 2^{\log_2 4} $.
3. Substitute into $ f(x) $ using the power rule $ (a^b)^c = a^{b \cdot c} $: $ f(x) = (2^{\log_2 4})^{(5x-1)} = 2^{(\log_2 4 \cdot (5x-1))} $.
4. Rewrite base 8 using logarithms: $ 8 = 2^{\log_2 8} $.
5. Substitute into $ g(x) $: $ g(x) = (2^{\log_2 8})^{(x/4)} = 2^{(\log_2 8 \cdot (x/4))} $.
6. This effectively sets up the equation $ 2^{2(5x-1)} = 2^{3(x/4)} $ for easy solving.

Question

The sales of a new product, in items per month, is modeled by the expression $225 + 500 \log_{10}(15t + 10)$, where $t$ represents the time since the product became available for purchase, in months. What is the number of items sold per month for time $t = 6$?

(A) $725$

(B) $1225$

(D) $5225$

▶️ Answer/Explanation

Detailed solution

Identify the given expression: $225 + 500 \log_{10}(15t + 10)$.
Substitute the given value $t = 6$ into the expression.
Calculate the inner term: $15(6) + 10 = 90 + 10 = 100$.
Evaluate the logarithm: $\log_{10}(100) = 2$, since $10^2 = 100$.
Multiply the logarithmic result by the coefficient: $500 \times 2 = 1000$.
Add the initial constant: $225 + 1000 = 1225$.
The number of items sold per month at $t = 6$ is $1225$.
Therefore, the correct option is (B).

Question

The function $j$ is given by $j(x) = 3 \cdot 4^{(x+2)}$. Which of the following is an equivalent form for $j(x)$?

(A) $j(x) = 48^x$

(B) $j(x) = 48 \cdot 4^x$

(D) $j(x) = 9 \cdot 16^x$

▶️ Answer/Explanation

Detailed solution

The given function is $j(x) = 3 \cdot 4^{(x+2)}$.
Apply the product rule of exponents: $a^{m+n} = a^m \cdot a^n$.
Rewrite the expression as $j(x) = 3 \cdot (4^x \cdot 4^2)$.
Calculate the constant value: $4^2 = 16$.
Substitute the value back: $j(x) = 3 \cdot 16 \cdot 4^x$.
Multiply the constants: $3 \cdot 16 = 48$.
The equivalent form is $j(x) = 48 \cdot 4^x$.
Therefore, the correct option is (B).

Question

Which of the following functions is an equivalent form of the function $p(x) = 4^{(-2x)}$?

(A) $p(x) = -(16)^x$

(B) $p(x) = (2)^x$

(D) $p(x) = \left(\frac{1}{16}\right)^x$

▶️ Answer/Explanation

Detailed solution

The given function is $p(x) = 4^{-2x}$.
Using the power of a power rule, $(a^m)^n = a^{m \cdot n}$, rewrite the expression as $(4^{-2})^x$.
Apply the negative exponent rule, $a^{-n} = \frac{1}{a^n}$, so $4^{-2} = \frac{1}{4^2}$.
Calculate the value of the base: $\frac{1}{4^2} = \frac{1}{16}$.
Substitute the base back into the function to get $p(x) = \left(\frac{1}{16}\right)^x$.
Comparing this result to the given options, we find it matches option (D).
Therefore, the equivalent form of the function is $p(x) = \left(\frac{1}{16}\right)^x$.

Question

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

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

(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).

AP Precalculus -2.4 Exponential Function Manipulation- MCQ Exam Style Questions - Effective Fall 2023