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

Explanation:

Step1: Analyze domain

The exponential - function $y = e^{u}$ is defined for all real values of $u$. For the function $f(x)=2 - e^{-x/2}$, there are no restrictions on the value of $x$. So the domain is all real numbers. Domain: $(-\infty,\infty)$

Step2: Analyze range

We know that the range of the basic exponential function $y = e^{x}$ is $(0,\infty)$. For the function $y = e^{-x/2}$, we can rewrite it as $y=\frac{1}{e^{x/2}}$, and its range is also $(0,\infty)$. Then, for $y=-e^{-x/2}$, the range is $(-\infty,0)$. Finally, for $f(x)=2 - e^{-x/2}$, we shift the graph of $y =-e^{-x/2}$ up by 2 units. So the range is $(-\infty,2)$. Range: $(-\infty,2)$

Step3: Analyze horizontal asymptote

As $x\rightarrow\infty$, $e^{-x/2}=\frac{1}{e^{x/2}}\rightarrow0$. Then $f(x)=2 - e^{-x/2}\rightarrow2$. So the horizontal asymptote is $y = 2$. Horizontal asymptote: $y = 2$

Answer:

Domain: $(-\infty,\infty)$ Range: $(-\infty,2)$ Horizontal asymptote: $y = 2$