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.

( h(x)=e^{x - 1}+1 )

graph ( h(x)=e^{x - 1}+1 ) and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.

Explanation:

Step1: Analyze horizontal shift

The parent function is \( f(x) = e^x \). For \( h(x) = e^{x - 1}+1 \), the \( x - 1 \) inside the exponent means we shift the graph of \( f(x) \) 1 unit to the right.

Step2: Analyze vertical shift

The \( +1 \) outside the exponential part means we shift the graph of \( f(x) \) (after the horizontal shift) 1 unit up.

Step3: Determine the asymptote

For the parent function \( f(x)=e^x \), the horizontal asymptote is \( y = 0 \). After shifting 1 unit up, the horizontal asymptote of \( h(x)=e^{x - 1}+1 \) is \( y=1 \) (since vertical shifts affect the horizontal asymptote in exponential functions of the form \( e^{x - h}+k \), the asymptote is \( y = k \)).

Step4: Domain and range

• Domain: For any exponential function of the form \( e^{x - h}+k \), the domain is all real numbers, so the domain of \( h(x) \) is \( (-\infty,\infty) \).
• Range: Since \( e^{x - 1}>0 \) for all real \( x \), then \( e^{x - 1}+1>1 \), so the range is \( (1,\infty) \).

Answer:

• Asymptote equation: \( y = 1 \) (dashed line)
• Domain: \( (-\infty,\infty) \)
• Range: \( (1,\infty) \)
• To graph \( h(x)=e^{x - 1}+1 \), start with the graph of \( y = e^x \), shift it 1 unit to the right and 1 unit up. The asymptote is \( y = 1 \) (dashed line).