Questions: Use the graph of y=e^x and transformations to sketch the exponential function f(x)=e^-x+5. Determine the domain and range. Also, determine the y-intercept, and find the equation of the horizontal asymptote.

Use the graph of y=e^x and transformations to sketch the exponential function f(x)=e^-x+5. Determine the domain and range. Also, determine the y-intercept, and find the equation of the horizontal asymptote.

Transcript text: Use the graph of $y=e^{x}$ and transformations to sketch the exponential function $f(x)=e^{-x}+5$. Determine the domain and range. Also, determine the $y$-intercept, and find the equation of the horizontal asymptote.

Solution

Solution Steps

Step 1: Determine the domain and range

The domain of $ f(x) = e^{-x} + 5 $ is all real numbers, $ (-\infty, \infty) $.

The range of $ f(x) = e^{-x} + 5 $ is $ (5, \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 $: \[ f(0) = e^{-0} + 5 = 1 + 5 = 6 \] So, the y-intercept is $ (0, 6) $.

Step 3: Find the equation of the horizontal asymptote

As $ x $ approaches infinity, $ e^{-x} $ approaches 0. Therefore, the horizontal asymptote is: \[ y = 5 \]

Final Answer

Domain: $ (-\infty, \infty) $
Range: $ (5, \infty) $
y-intercept: $ (0, 6) $
Horizontal asymptote: $ y = 5 $

{"axisType": 3, "coordSystem": {"xmin": -3, "xmax": 3, "ymin": 4, "ymax": 10}, "commands": ["y = ex", "y = e(-x) + 5"], "latex_expressions": ["$y = e^{x}$", "$f(x) = e^{-x} + 5$"]}

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!