Questions: Use the graph of y=e^x and transformations to sketch the exponential function f(x)=e^(-x)-2. 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)-2. 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 $f(x)=e^{-x}-2$. 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} - 2 $ is all real numbers, $ (-\infty, \infty) $.

The range of $ f(x) = e^{-x} - 2 $ is $ (-2, \infty) $ because $ e^{-x} $ is always positive and the minimum value of $ e^{-x} $ is 0, so the minimum value of $ f(x) $ is $ -2 $.

Step 2: Find the y-intercept

To find the y-intercept, set $ x = 0 $: \[ f(0) = e^{-0} - 2 = 1 - 2 = -1 \] So, the y-intercept is $ (0, -1) $.

Step 3: Find the equation of the horizontal asymptote

As $ x \to \infty $, $ e^{-x} \to 0 $. Therefore, $ f(x) \to -2 $. The equation of the horizontal asymptote is $ y = -2 $.

Final Answer

Domain: $ (-\infty, \infty) $
Range: $ (-2, \infty) $
y-intercept: $ (0, -1) $
Horizontal asymptote: $ y = -2 $

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