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} )
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} ).
( (-infty,infty) )
(type your answer in interval notation.)
observe the graph and find the range of ( g(x)=e^{x - 2} ).
(type your answer in interval notation.)

Explanation:

Step1: Recall properties of exponential functions

The parent function is \( f(x) = e^x \), which has a horizontal asymptote at \( y = 0 \), domain \( (-\infty, \infty) \), and range \( (0, \infty) \). For the function \( g(x)=e^{x - 2} \), this is a horizontal shift (right by 2 units) of the parent function \( f(x)=e^x \). Horizontal shifts do not affect the horizontal asymptote or the range in terms of the lower bound, and also do not restrict the domain.

Step2: Determine the range of \( g(x) \)

Since the parent function \( e^x \) has a range of \( (0, \infty) \), and a horizontal shift ( \( x-2 \) instead of \( x \)) does not change the output values' lower bound (the function still approaches 0 as \( x \to -\infty \) and increases without bound as \( x \to \infty \)), the range of \( g(x)=e^{x - 2} \) is also \( (0, \infty) \).

Answer:

\((0, \infty)\)