use the function below to answer the following questions\n g(x)=e^{x - 2}\n(a) use transformations of the…

Explanation:

Step1: Analyze graph transformation

The function $g(x)=e^{x - 2}$ is a horizontal - shift of $y = e^{x}$. The general form of a horizontal shift of a function $y = f(x)$ to $y=f(x - h)$ is a shift of $h$ units to the right. For $y = e^{x}$ shifted to $g(x)=e^{x - 2}$, it is shifted 2 units to the right.

Step2: Find domain

The exponential function $y = e^{x}$ has a domain of all real numbers. Since $g(x)=e^{x - 2}$ is also an exponential - type function, the domain of $g(x)$ is $(-\infty,\infty)$.

Step3: Find range

The exponential function $y = e^{x}$ has a range of $(0,\infty)$. Shifting the graph horizontally does not affect the range. So the range of $g(x)=e^{x - 2}$ is $(0,\infty)$.

Step4: Find asymptote

The exponential function $y = e^{x}$ has a horizontal asymptote at $y = 0$. A horizontal shift does not change the horizontal asymptote. So the equation of the horizontal asymptote of $g(x)=e^{x - 2}$ is $y = 0$.

Answer:

(a) Shift the graph of $y = e^{x}$ 2 units to the right. (b) Domain: $(-\infty,\infty)$; Range: $(0,\infty)$ (c) $y = 0$