The exponential function $f$ is graphed in the $x y$ -plane. As $x$ increases by $1, y$ increases by a factor of $3$ . Which of the following could be $f$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $f(x)=\left(\frac{1}{3}\right)^{x}$ $\newline$ (B) $f(x)=\left(\frac{1}{3}\right)^{x}+3$ $\newline$ (C) $f(x)=3^{x}+2$ $\newline$ (D) $f(x)=2(3)^{x}$

View step-by-step help

Home

Math Problems

Precalculus

Find equations of tangent lines using limits

Full solution

Q. The exponential function

f

is graphed in the

x y

-plane. As

x

increases by

1, y

increases by a factor of

3

. Which of the following could be

f

\newline

Choose

1

answer:

\newline

(A)

f(x)=\left(\frac{1}{3}\right)^{x}

\newline

(B)

f(x)=\left(\frac{1}{3}\right)^{x}+3

\newline

(C)

f(x)=3^{x}+2

\newline

(D)

f(x)=2(3)^{x}

Identify Base of Exponential Function: We are given that as $x$ increases by $1$ , $y$ increases by a factor of $3$ . This means that the base of the exponential function must be $3$ . We can eliminate any options that do not have a base of $3$ .
Eliminate Options with Incorrect Bases: Option (A) $f(x) = (\frac{1}{3})^x$ has a base of $\frac{1}{3}$ , which would mean that $y$ decreases as $x$ increases, because $\frac{1}{3}$ is less than $1$ . This does not match the description given in the problem.
Analyze Option (A): Option (B) $f(x) = (\frac{1}{3})^x + 3$ also has a base of $\frac{1}{3}$ , which, as previously stated, would mean that $y$ decreases as $x$ increases. The addition of $3$ does not change the fact that the base is still $\frac{1}{3}$ , so this option is also incorrect.
Analyze Option (B): Option (C) $f(x) = 3^x + 2$ has a base of $3$ , which fits the description of the problem. However, we need to check if the ＂+ $2$ ＂ affects the property that $y$ increases by a factor of $3$ as $x$ increases by $1$ .
Analyze Option (C): The ＂+ $2$ ＂ in option (C) is a vertical shift of the graph of the function $f(x) = 3^x$ . It does not affect the factor by which $y$ increases when $x$ increases by $1$ . The factor of increase is still determined by the base of the exponential, which is $3$ . Therefore, option (C) could represent the function described in the problem.
Analyze Option (D): Option (D) $f(x) = 2(3)^x$ has a base of $3$ , which is correct. However, the factor of $2$ in front of the base means that $y$ is not just increasing by a factor of $3$ as $x$ increases by $1$ , but rather by a factor of $2 \times 3$ . This does not match the description given in the problem.