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 graphs to determine each functions domain and range. if applicable, use a graphing utility to confirm the hand - drawn graphs.
( g(x)=e^{x}+5 )
the graph of ( f(x)=e^{x} ) should be shifted 5 units downward
the graph of ( f(x)=e^{x} ) should be shifted 5 units to the left
the graph of ( f(x)=e^{x} ) should be shifted 5 units upward
graph ( g(x) ) and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.

Explanation:

Step1: Analyze the transformation

For a function \( y = f(x)+k \), if \( k>0 \), the graph of \( y = f(x) \) is shifted \( k \) units upward. Here, \( f(x)=e^{x} \) and \( g(x)=e^{x}+5 \), so \( k = 5>0 \). Thus, the graph of \( f(x)=e^{x} \) is shifted 5 units upward.

Step2: Find the asymptote

The parent function \( f(x)=e^{x} \) has a horizontal asymptote \( y = 0 \). When we shift the graph upward by 5 units, the horizontal asymptote of \( g(x)=e^{x}+5 \) also shifts upward by 5 units. So the equation of the asymptote is \( y=5 \).

Step3: Determine domain and range

• Domain: The exponential function \( e^{x} \) is defined for all real numbers, and shifting the graph vertically does not affect the domain. So the domain of \( g(x) \) is \( (-\infty,\infty) \).
• Range: The range of \( f(x)=e^{x} \) is \( (0,\infty) \). When we shift the graph upward by 5 units, each \( y \)-value of \( f(x) \) increases by 5. So the range of \( g(x)=e^{x}+5 \) is \( (5,\infty) \).

Answer:

• Transformation: The graph of \( f(x) = e^{x} \) is shifted 5 units upward.
• Asymptote equation: \( y = 5 \)
• Domain: \( (-\infty, \infty) \)
• Range: \( (5, \infty) \)