Question

use transformations of the graph of $f(x)=e^{x}$ to graph the given function. be sure to give the equation of the asymptote. use the graphs to determine each functions domain and range. if applicable, use a graphing utility to confirm the hand - drawn graphs.
$h(x)=e^{x + 2}-3$
which transformations are needed to graph the function $h(x)=e^{x + 2}-3$? choose the correct answer below.
a. the graph of $f(x)=e^{x}$ should be shifted to the right by 2 units and shift $f(x)$ upward by 3 units.
b. the graph of $f(x)=e^{x}$ should be shifted to the left by 2 units and shift $f(x)$ upward by 3 units.
c. the graph of $f(x)=e^{x}$ should be shifted to the left by 2 units and shift $f(x)$ downward by 3 units.
d. the graph of $f(x)=e^{x}$ should be shifted to the right by 2 units and shift $f(x)$ downward by 3 units.
graph $h(x)=e^{x + 2}-3$ and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.
find the equation of the asymptote for $h(x)=e^{x + 2}-3$ using the graph.
$y=-3$
(type an equation.)
observe the graph and find the domain of $h(x)=e^{x + 2}-3$
(type your answer in interval notation.)

Explanation:

Step1: Determine the domain of an exponential function

For any exponential function of the form \(y = a\cdot e^{bx + c}+d\), the domain is all real numbers. The function \(h(x)=e^{x + 2}-3\) is an exponential function. There is no restriction on the value of \(x\) for which the function \(h(x)=e^{x+2}-3\) is defined.

Answer:

\((-\infty,\infty)\)