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

Explanation:

Step1: Recall the parent function \( y = e^x \)

The parent function \( y=e^x \) has a horizontal asymptote \( y = 0 \), domain \( (-\infty,\infty) \), and range \( (0,\infty) \).

Step2: Analyze the transformation for \( h(x)=e^{3x}-6 \)

For the function \( h(x)=e^{3x}-6 \), we consider the general form of an exponential function \( y = a\cdot e^{bx}+c \). Here \( a = 1 \), \( b=3 \), \( c=-6 \).
The horizontal asymptote of \( y = e^{3x}-6 \) is found by considering the limit as \( x\to-\infty \). Since \( \lim_{x\to-\infty}e^{3x}=0 \), the horizontal asymptote is \( y=-6 \).
The domain of an exponential function \( y = e^{u(x)}+k \) (where \( u(x)=3x \) in our case) is all real numbers because the exponential function \( e^{3x} \) is defined for all \( x\in R \). So, the domain of \( h(x) \) is \( (-\infty,\infty) \).
For the range, we know that \( e^{3x}>0 \) for all \( x\in R \). Then \( e^{3x}-6>-6 \). So the range is \( (-6,\infty) \).

Answer:

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