begin with the graph of y = e^x. use transformations to graph the function below. then determine its domain…

Explanation:

Step1: Analyze the parent - function and transformations

The parent function is $y = e^{x}$. The function $f(x)=-8 - e^{-x}$ can be obtained from $y = e^{x}$ through the following transformations: First, reflect $y = e^{x}$ about the $y$-axis to get $y = e^{-x}$, then reflect it about the $x$-axis to get $y=-e^{-x}$, and finally shift it down 8 units to get $y=-8 - e^{-x}$.

Step2: Determine the domain

The exponential function $y = e^{-x}$ is defined for all real - valued $x$. So, the domain of $f(x)=-8 - e^{-x}$ is all real numbers. In interval notation, the domain is $(-\infty,\infty)$.

Step3: Determine the range

We know that the range of $y = e^{-x}$ is $(0,\infty)$. After reflecting it about the $x$-axis, the range of $y=-e^{-x}$ is $(-\infty,0)$. Then, after shifting it down 8 units, the range of $y=-8 - e^{-x}$ is $(-\infty, - 8)$. In interval notation, the range is $(-\infty,-8)$.

Step4: Determine the horizontal asymptote

As $x\to\infty$, $e^{-x}=\frac{1}{e^{x}}\to0$. So, $f(x)=-8 - e^{-x}\to - 8$. The horizontal asymptote is $y = - 8$.

Answer:

Domain: $(-\infty,\infty)$ Range: $(-\infty,-8)$ Horizontal asymptote: $y=-8$