Question

use transformations of the graph of ( f(x)=e^{x} ) to graph the given function. be sure to the give equations of the asymptotes. use the graphs to determine each functions domain and range. if applicable, use a graphing utility to confirm the hand - drawn graphs.
( g(x)=e^{x - 2} )
graph ( g(x) ) 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 ( g(x)=e^{x - 2} ) using the graph.
(type an equation.)

Explanation:

Step1: Recall transformation of exponential functions

For the parent function \( f(x) = e^x \), the horizontal asymptote is \( y = 0 \) (since as \( x \to -\infty \), \( e^x \to 0 \)). When we have a function of the form \( g(x)=e^{x - h} \), it is a horizontal shift of the parent function \( f(x)=e^x \). A horizontal shift does not change the horizontal asymptote of the exponential function.

Step2: Determine the asymptote of \( g(x)=e^{x - 2} \)

Since \( g(x)=e^{x - 2} \) is a horizontal shift (2 units to the right) of \( f(x)=e^x \), the horizontal asymptote remains the same as that of \( f(x) \), which is \( y = 0 \).

Answer:

\( y = 0 \)