which is the graph of $g(x)=2^{x - 1}+3$?

Explanation:

Step1: Analyze the parent - function

The parent - function of $g(x)=2^{x - 1}+3$ is $y = 2^{x}$. The general form of an exponential function is $y=a\cdot b^{x - h}+k$, where $(h,k)$ is the horizontal and vertical shift of the graph. For $g(x)=2^{x - 1}+3$, $a = 1$, $b = 2$, $h = 1$, and $k = 3$.

Step2: Consider the horizontal shift

The horizontal shift is given by $h$. Since $h = 1$, the graph of $y = 2^{x}$ is shifted 1 unit to the right.

Step3: Consider the vertical shift

The vertical shift is given by $k$. Since $k = 3$, the graph of $y = 2^{x-1}$ is shifted 3 units up.

Step4: Find the y - intercept

To find the y - intercept, set $x = 0$ in $g(x)$: [ \begin{align*} g(0)&=2^{0 - 1}+3\ &=2^{-1}+3\ &=\frac{1}{2}+3\ &=\frac{1 + 6}{2}=\frac{7}{2}=3.5 \end{align*} ]

Step5: Analyze the asymptote

The horizontal asymptote of the parent - function $y = 2^{x}$ is $y = 0$. After the vertical shift of 3 units up, the horizontal asymptote of $g(x)=2^{x - 1}+3$ is $y = 3$.

The graph that has a horizontal asymptote at $y = 3$, passes through the point $(0,3.5)$ and has the general shape of an exponential growth function (since $b=2>1$) is the correct graph.

Answer:

The graph with a horizontal asymptote at $y = 3$, passing through $(0,3.5)$ and having an exponential - growth shape.