Question

begin with the graph of ( y = e ^ { x } ) and use transformations to graph the following function. determine the domain, range, horizontal asymptote, and y - intercept of the function.
( f ( x ) = e ^ { - x } )

Explanation:

Step1: Analyze the transformation

The function \(y = e^{-x}\) is a reflection of \(y = e^{x}\) about the \(y\)-axis.

Step2: Determine the domain

For any real - valued \(x\), the function \(f(x)=e^{-x}=\frac{1}{e^{x}}\) is defined. So, the domain is \((-\infty,\infty)\)

Step3: Determine the range

Since \(e^{-x}>0\) for all \(x\in R\), and as \(x
ightarrow\infty\), \(e^{-x}
ightarrow0\) and as \(x
ightarrow-\infty\), \(e^{-x}
ightarrow\infty\). So the range is \((0,\infty)\)

Step4: Determine the horizontal asymptote

As \(x
ightarrow\infty\), \(y = e^{-x}
ightarrow0\). So the horizontal asymptote is \(y = 0\)

Step5: Determine the \(y\)-intercept

To find the \(y\)-intercept, set \(x = 0\). Then \(f(0)=e^{-0}=1\)

Answer:

• Domain: \((-\infty,\infty)\)
• Range: \((0,\infty)\)
• Horizontal asymptote: \(y = 0\)
• \(y\)-intercept: \((0,1)\)