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^{2 x}+6 )
graph ( h(x) ) and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.

Explanation:

Step1: Analyze the transformation of the function

The parent function is \(y = e^{x}\). For the function \(h(x)=e^{2x}+6\), compared with \(y = e^{x}\), there is a horizontal compression by a factor of \(\frac{1}{2}\) (due to the \(2x\) inside the exponent) and a vertical shift up by \(6\) units.

The general form of an exponential function is \(y = a\cdot e^{bx}+c\). For the function \(y = e^{x}\), the asymptote is \(y = 0\).

Step2: Find the asymptote

For the function \(h(x)=e^{2x}+6\), using the transformation rules. When we shift the function \(y = e^{x}\) up by \(6\) units, the equation of the asymptote changes. If \(y = f(x)\) has an asymptote \(y = k\), then \(y=f(x)+m\) has an asymptote \(y=k + m\). Since the asymptote of \(y = e^{x}\) is \(y = 0\), for \(h(x)=e^{2x}+6\), the asymptote is \(y = 6\)

Step3: Determine the domain

The domain of an exponential function \(y = a\cdot e^{bx}+c\) (where \(a
eq0\), \(b
eq0\)) is all real numbers. For \(h(x)=e^{2x}+6\), since we can substitute any real - value of \(x\) into the function \(h(x)\), the domain is \((-\infty,\infty)\)

Step4: Determine the range

We know that \(e^{2x}>0\) for all \(x\in R\). Then \(h(x)=e^{2x}+6>0 + 6\). So the range is \((6,\infty)\)

Answer:

Asymptote: \(y = 6\); Domain: \((-\infty,\infty)\); Range: \((6,\infty)\)