Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![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 \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42dff072-7d42-411c-b912-034f8f72c397%2Fd0c0251b-1e10-4835-864c-18334785c88f%2F7qlayar_processed.png&w=3840&q=75)
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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