Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Graph the function: \[ f(x) = \begin{cases} -3 & \text{if } -10 < x \leq -3 \\ 4 - x^2 & \text{if } -3 \leq x \leq 2 \\ \frac{1}{2}x - 2 & \text{if } x > 2 \end{cases} \] Explanation: 1. **For \( -10 < x \leq -3 \):** The function \( f(x) = -3 \) represents a horizontal line at \( y = -3 \). 2. **For \( -3 \leq x \leq 2 \):** The function \( f(x) = 4 - x^2 \) is a downward-opening parabola with vertex at \( (0, 4) \). 3. **For \( x > 2 \):** The function \( f(x) = \frac{1}{2}x - 2 \) is a line with a slope of \( \frac{1}{2} \) and y-intercept at \(-2\). The piecewise function consists of linear and quadratic components, with distinct behaviors across different intervals of \( x \).
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Transcribed Image Text:Graph the function: \[ f(x) = \begin{cases} -3 & \text{if } -10 < x \leq -3 \\ 4 - x^2 & \text{if } -3 \leq x \leq 2 \\ \frac{1}{2}x - 2 & \text{if } x > 2 \end{cases} \] Explanation: 1. **For \( -10 < x \leq -3 \):** The function \( f(x) = -3 \) represents a horizontal line at \( y = -3 \). 2. **For \( -3 \leq x \leq 2 \):** The function \( f(x) = 4 - x^2 \) is a downward-opening parabola with vertex at \( (0, 4) \). 3. **For \( x > 2 \):** The function \( f(x) = \frac{1}{2}x - 2 \) is a line with a slope of \( \frac{1}{2} \) and y-intercept at \(-2\). The piecewise function consists of linear and quadratic components, with distinct behaviors across different intervals of \( x \).
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