begin with the graph of y = e^x and use transformations to graph the function. determine the domain, range…

Explanation:

Step1: Analyze the exponential - function transformation

We know that the general form of an exponential function is $y = a\cdot e^{bx}+c$. For the function $y = e^x$, the given function $f(x)=4 - e^{-x/2}$ can be analyzed in terms of transformations. The domain of the exponential function $y = e^x$ is $(-\infty,\infty)$, and since there are no restrictions on the value of $x$ for the function $f(x)=4 - e^{-x/2}$, the domain remains $(-\infty,\infty)$. To find the range, we consider the behavior of the exponential part $y = e^{-x/2}$.

Step2: Determine the range of $y = e^{-x/2}$

The function $y = e^{-x/2}=\left(e^{-1/2}\right)^x=\left(\frac{1}{\sqrt{e}}\right)^x$. The range of the exponential function $y = e^{-x/2}$ is $(0,\infty)$ because for any real - valued $x$, $e^{-x/2}>0$.

Step3: Find the range of $f(x)=4 - e^{-x/2}$

Let $t = e^{-x/2}$. Then $f(x)=4 - t$. Since $t\in(0,\infty)$, when we consider $y = 4 - t$, as $t$ approaches $0$, $y$ approaches $4$, and as $t$ approaches $\infty$, $y$ approaches $-\infty$. So the range of $f(x)$ is $(-\infty,4)$.

Step4: Find the horizontal asymptote

As $x\rightarrow\infty$, $e^{-x/2}\rightarrow0$. So $f(x)=4 - e^{-x/2}\rightarrow4$. The horizontal asymptote is $y = 4$.

Answer:

Range: $(-\infty,4)$