Questions: Use the graph of y=e^x and transformations to sketch the exponential function f(x)=e^-x-4. 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-4. Determine the domain and range. Also, determine the y-intercept, and find the equation of the horizontal asymptote.

Transcript text: Use the graph of $y=e^{\mathrm{x}}$ and transformations to sketch the exponential function $\mathrm{f}(\mathrm{x})=e^{-\mathrm{x}}-4$. Determine the domain and range. Also, determine the $y$-intercept, and find the equation of the horizontal asymptote.

Solution

Solution Steps

Step 1: Identify the Base Function

The base function given is $ y = e^x $.

Step 2: Apply the Transformation

The given function is $ f(x) = e^{-x} - 4 $. This involves two transformations:

Reflecting $ e^x $ across the y-axis to get $ e^{-x} $.
Shifting the graph of $ e^{-x} $ downward by 4 units.

Step 3: Determine the Domain

The domain of $ f(x) = e^{-x} - 4 $ is all real numbers, $ (-\infty, \infty) $, because exponential functions are defined for all real numbers.

Step 4: Determine the Range

The range of $ f(x) = e^{-x} - 4 $ is $ (-4, \infty) $. This is because $ e^{-x} $ is always positive, and subtracting 4 shifts the entire graph downward by 4 units.

Step 5: Find the y-Intercept

To find the y-intercept, set $ x = 0 $: \[ f(0) = e^{-0} - 4 = 1 - 4 = -3 \] So, the y-intercept is $ (0, -3) $.

Step 6: Determine the Horizontal Asymptote

The horizontal asymptote of $ f(x) = e^{-x} - 4 $ is $ y = -4 $. This is because as $ x $ approaches infinity, $ e^{-x} $ approaches 0, and thus $ f(x) $ approaches -4.