Answer
Refer to the image below for the graph. The graph of $g(x)$ is green, the graph of the parent function is red.
Domain: $(-\infty,\infty)$
Range: $(0,\infty)$
Horizontal asymptote: $y=0$
Work Step by Step
$g(x)=2^{x-3}$
RECALL:
The graph of the function $f(x)=a^{x−h}$ involves a horizontal shift of $|h|$ units (to the right when $h$ is positive, to the left when $h$ is negative) of the parent function $y=a^x$.
APPLY:
The given function $g(x)$'s parent function is $y=2^x$. The appropriate value of $h$, therefore, is $3$. Since $h=3$, $g(x)$'s graph involves a 3-unit horizontal shift to the right of the parent function.
Thus, to graph the given function, perform the following steps:
(1) Graph the parent function $y=2^x$ by creating a table of values then connecting the points using a smooth curve.
(refer to the attached table below, the graph is attached in the answer part above)
Thus, to graph the given function, perform the following steps:
(1) Graph the parent function $y=2^x$ by creating a table of values then connecting the points using a smooth curve (refer to the attached table below).
(2) Draw the graph of $g(x)$ by shifting the points of the parent function 3 units to the right (refer to the attached image in the answer part above for the graph).
Since $g(x)$ is an exponential function, the domain is all real numbers. Since the base of the exponent ($2$) is a positive number, the range of $g(x)$ is also all positive numbers.
The horizontal asymptote of a valid exponential function $y=a^x+b$ is $y=b$. Since the function $g(x)$ has no constant ($b=0$), the horizontal asymptote is $y=0$.
