Questions: Use the graph of y=e^x and transformations to sketch the exponential function f(x)=e^(-x)+6. Determine the domain and range. Also, determine the y-intercept, and find the equation of the horizontal asymptote. Use the graphing tool to graph the function. Click to enlarge Cljaph

Use the graph of y=e^x and transformations to sketch the exponential function f(x)=e^(-x)+6. Determine the domain and range. Also, determine the y-intercept, and find the equation of the horizontal asymptote.

Use the graphing tool to graph the function.
Click to enlarge
Cljaph

Transcript text: mylab.pearson.com Section 2.5 Applications Decay Applications Decay Question 2, AR.9.7 Part 1 of 5 Use the graph of $\mathrm{y}=e^{\mathrm{x}}$ and transformations to sketch the exponential function $\mathrm{f}(\mathrm{x})=e^{-\mathrm{x}}+6$. Determine the domain and range. Also, determine the $y$-intercept, and find the equation of the horizontal asymptote. Use the graphing tool to graph the function. \begin{tabular}{|c|c|} \hline Click to \\ enlarge \\ Cljaph \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Determine the domain and range

The domain of $ f(x) = e^{-x} + 6 $ is all real numbers, $ (-\infty, \infty) $.

The range of $ f(x) = e^{-x} + 6 $ is $ (6, \infty) $ because $ e^{-x} $ is always positive and approaches 0 as $ x $ approaches infinity.

Step 2: Determine the y-intercept

To find the y-intercept, set $ x = 0 $: \[ f(0) = e^{-0} + 6 = 1 + 6 = 7 \] So, the y-intercept is $ (0, 7) $.

Step 3: Find the equation of the horizontal asymptote

As $ x $ approaches infinity, $ e^{-x} $ approaches 0. Therefore, the horizontal asymptote is $ y = 6 $.

Final Answer

Domain: $ (-\infty, \infty) $
Range: $ (6, \infty) $
y-intercept: $ (0, 7) $
Horizontal asymptote: $ y = 6 $

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