Question

use transformations of the graph of ( f(x)=e^{x} ) to graph the given function. p 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.
( y = 0 )
(type an equation.)
observe the graph and find the domain of ( g(x)=e^{x - 2} ).
(type your answer in interval notation.)

Explanation:

Step1: Recall domain of exponential function

The parent function \( f(x) = e^x \) has a domain of all real numbers, \( (-\infty, \infty) \), because the exponential function is defined for any real - valued input \( x \).

Step2: Analyze the transformation for \( g(x)=e^{x - 2} \)

The function \( g(x)=e^{x - 2} \) is a horizontal shift of the parent function \( f(x)=e^x \). A horizontal shift of the form \( y = f(x - h) \) (in this case, \( h = 2 \)) does not affect the domain of the function. The exponential function \( e^{u} \), where \( u=x - 2 \), is defined for all real values of \( x \) (since \( x-2 \) is a real number for any real \( x \)). So, the domain of \( g(x)=e^{x - 2} \) is the set of all real numbers.

Answer:

\((-\infty, \infty)\)