Questions: Use the graph of (y=2^x) and transformations to sketch the exponential function. Determine the domain and range. Also, determine the (y)-intercept, and find the equation of the horizontal asymptote. [f(x)=-2^x-2-2] Use the graphing tool to graph the function. What is the domain of (f(x))? (Type your answer in interval notation.)

Use the graph of (y=2^x) and transformations to sketch the exponential function. Determine the domain and range. Also, determine the (y)-intercept, and find the equation of the horizontal asymptote. [f(x)=-2^x-2-2] Use the graphing tool to graph the function. What is the domain of (f(x))? (Type your answer in interval notation.)
Transcript text: mylab.pearson.com LTI Launch Skills Review Homework Question 3, AR.9.3 Part 2 of 5 Use the graph of $y=2^{x}$ and transformations to sketch the exponential function. Determine the domain and range. Also, determine the $y$-intercept, and find the equation of the horizontal asymptote. \[ f(x)=-2^{x-2}-2 \] Use the graphing tool to graph the function. $\square$ What is the domain of $f(x)$ ? $\square$ (Type yots answer in interval notation.) example Ask my instructor
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Solution

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Solution Steps

Step 1: Identify the given function and its transformations

The given function is: \[ f(x) = -2^{x-2} - 2 \]

Step 2: Determine the domain of the function

The domain of the function \( f(x) = -2^{x-2} - 2 \) is all real numbers, since the exponential function is defined for all \( x \in \mathbb{R} \).

Step 3: Determine the range of the function

The range of the function \( f(x) = -2^{x-2} - 2 \) is all real numbers less than \(-2\), since the exponential function \( 2^{x-2} \) is always positive and the negative sign in front of it makes it always negative, and then subtracting 2 shifts it down by 2 units.

Step 4: Determine the y-intercept

To find the y-intercept, set \( x = 0 \): \[ f(0) = -2^{0-2} - 2 = -2^{-2} - 2 = -\frac{1}{4} - 2 = -2.25 \]

Step 5: Find the equation of the horizontal asymptote

The horizontal asymptote of the function \( f(x) = -2^{x-2} - 2 \) is \( y = -2 \).

Final Answer

  • Domain: \( (-\infty, \infty) \)
  • Range: \( (-\infty, -2) \)
  • y-intercept: \( -2.25 \)
  • Horizontal asymptote: \( y = -2 \)

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