use transformations of the graph of $f(x)=e^{x}$ to graph the given function. be sure to give the equation…

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.\n$h(x)=e^{x + 2}-3$\nshifted to the right by 2 units and shift $f(x)$ upward by 3 units.\nthe graph of $f(x)=e^{x}$ should be shifted to the left by 2 units and shift $f(x)$ upward by 3 units.\nthe graph of $f(x)=e^{x}$ should be shifted to the left by 2 units and shift $f(x)$ downward by 3 units.\nthe graph of $f(x)=e^{x}$ should be shifted to the right by 2 units and shift $f(x)$ downward by 3 units.\ngraph $h(x)=e^{x + 2}-3$ and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.\nfind the equation of the asymptote for $h(x)=e^{x + 2}-3$ using the graph.\n$y=-3$\n(type an equation.)\nobserve the graph and find the domain of $h(x)=e^{x + 2}-3$.\n$(-\\infty,\\infty)$\n(type your answer in interval notation.)\nobserve the graph and find the range of $h(x)=e^{x + 2}-3$.\n(type your answer in interval notation.)

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.\n$h(x)=e^{x + 2}-3$\nshifted to the right by 2 units and shift $f(x)$ upward by 3 units.\nthe graph of $f(x)=e^{x}$ should be shifted to the left by 2 units and shift $f(x)$ upward by 3 units.\nthe graph of $f(x)=e^{x}$ should be shifted to the left by 2 units and shift $f(x)$ downward by 3 units.\nthe graph of $f(x)=e^{x}$ should be shifted to the right by 2 units and shift $f(x)$ downward by 3 units.\ngraph $h(x)=e^{x + 2}-3$ and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.\nfind the equation of the asymptote for $h(x)=e^{x + 2}-3$ using the graph.\n$y=-3$\n(type an equation.)\nobserve the graph and find the domain of $h(x)=e^{x + 2}-3$.\n$(-\\infty,\\infty)$\n(type your answer in interval notation.)\nobserve the graph and find the range of $h(x)=e^{x + 2}-3$.\n(type your answer in interval notation.)

Answer

Explanation:

Step1: Analyze the transformation of the exponential function

For the function (y = e^{x + 2}-3), compared with the parent function (y = e^{x}), according to the transformation rule of the function (y = f(x + a)+b). When (a>0), the graph of (y = f(x)) is shifted to the left by (a) units; when (b<0), the graph of (y = f(x)) is shifted downward by (|b|) units. Here (a = 2) and (b=-3), so the graph of (y = e^{x}) is shifted to the left by 2 units and shifted downward by 3 units.

Step2: Determine the range of the function

The range of the parent - function (y = e^{x}) is ((0,\infty)). For the function (y=e^{x + 2}-3), let (t=x + 2), then (y = e^{t}-3). Since (e^{t}>0) for all real (t), then (e^{t}-3>-3).

Answer:

The range of (h(x)=e^{x + 2}-3) is ((-3,\infty))