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 graph to determine the functions domain and range. ( h(x)=e^{3 x}-3 ) graph ( h(x) ) and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function. click to enlarge graph

Explanation:

Step1: Analyze the transformation of the function

The parent function is \(y = e^{x}\). For the function \(h(x)=e^{3x}-3\), first, there is a horizontal compression by a factor of \(\frac{1}{3}\) (since \(y = f(ax)\) with \(a = 3\) for \(y=e^{x}\) to \(y = e^{3x}\)). Then, there is a vertical translation down by 3 units (since \(y=f(x)+k\) with \(k=- 3\)).

Step2: Find the asymptote

The asymptote of \(y = e^{x}\) is \(y = 0\). After the vertical translation \(y=e^{3x}-3\), the asymptote is \(y=-3\).

Step3: Determine the domain

The domain of \(y = e^{x}\) is \((-\infty,\infty)\). Since the transformations (horizontal compression and vertical translation) do not restrict the input values of \(x\), the domain of \(h(x)=e^{3x}-3\) is \((-\infty,\infty)\).

Step4: Determine the range

The range of \(y = e^{x}\) is \((0,\infty)\). After the vertical translation \(y = e^{3x}-3\), we subtract 3 from each output of \(y = e^{3x}\). So the range of \(h(x)\) is \((-3,\infty)\).

Answer:

• Asymptote: \(y = - 3\)
• Domain: \((-\infty,\infty)\)
• Range: \((-3,\infty)\)