Question $12$ $\newline$ The function $y=g(x)$ where $g(x)=3\left(\frac{1}{2}\right)^{-x}+2$ is graphed in the $x y$ -plane. Which of the following is a true statement? $\newline$ a. The graph of function $g$ is always increasing. $\newline$ b. The $y$ -intercept of the graph of function $g$ is $(0,2)$ . $\newline$ c. The $x$ -intercept of the graph of function $g$ is $(0,3)$ . $\newline$ d. Function $g$ is symmetric with respect to the $y$ -axis. $\newline$ A $\newline$ B $\newline$ c $\newline$ D

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Q. Question

12

\newline

The function

y=g(x)

where

g(x)=3\left(\frac{1}{2}\right)^{-x}+2

is graphed in the

x y

-plane. Which of the following is a true statement?

\newline

a. The graph of function

g

is always increasing.

\newline

b. The

y

-intercept of the graph of function

g

(0,2)

\newline

c. The

x

-intercept of the graph of function

g

(0,3)

\newline

d. Function

g

is symmetric with respect to the

y

-axis.

\newline

\newline

\newline

\newline

Rephrasing the Question: Let's rephrase the ＂Which statement about the graph of the function $g(x) = 3(\frac{1}{2})^{-x} + 2$ is true?＂
Determining Increasing Trend: To determine if the graph of function $g$ is always increasing, we need to look at the exponent of the term $(\frac{1}{2})^{-x}$ . Since the base $(\frac{1}{2})$ is between $0$ and $1$ , and the exponent is negative, this term represents exponential growth. Therefore, as $x$ increases, $(\frac{1}{2})^{-x}$ increases, making the function $g(x)$ increase as well. This suggests that the graph of function $g$ is always increasing.
Finding Y-Intercept: To find the y-intercept of the graph of function $g$ , we set $x$ to $0$ and calculate $g(0)$ . The y-intercept is the point where the graph crosses the y-axis, which occurs when $x = 0$ . $g(0) = 3(\frac{1}{2})^{-0} + 2 = 3(1) + 2 = 3 + 2 = 5$ The y-intercept of the graph of function $g$ is $(0, 5)$ , not $(0, 2)$ . Therefore, option b is incorrect.
Finding X-Intercept: To find the x-intercept of the graph of function $g$ , we set $g(x)$ to $0$ and solve for $x$ . The x-intercept is the point where the graph crosses the x-axis, which occurs when $y = 0$ . $0 = 3\left(\frac{1}{2}\right)^{-x} + 2$ Subtracting $2$ from both sides gives us: $-2 = 3\left(\frac{1}{2}\right)^{-x}$ This equation suggests that $\left(\frac{1}{2}\right)^{-x}$ would have to be negative for the equality to hold, which is impossible since any positive number raised to any power is positive. Therefore, there is no x-intercept, and option c is incorrect.
Checking Symmetry: To determine if function $g$ is symmetric with respect to the y-axis, we need to check if $g(-x) = g(x)$ for all $x$ . If this is true, then the function is even and symmetric with respect to the y-axis. $\newline$ $g(-x) = 3(\frac{1}{2})^{-(-x)} + 2 = 3(\frac{1}{2})^x + 2$ $\newline$ Since $g(-x)$ does not equal $g(x)$ , the function is not symmetric with respect to the y-axis, and option d is incorrect.
Correct Statement: Based on the analysis, the correct statement about the graph of the function $g(x)$ is that it is always increasing. Therefore, option $a$ is the correct answer.