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
![**Find the intervals on which the function is continuous.**
\[ \text{45)} \quad f(x) = \begin{cases}
-x^2 + 1, & -1 \leq x < 0 \\
4x, & 0 < x < 1 \\
-3, & x = 1 \\
-4x + 8, & 1 < x < 3 \\
4, & 3 < x < 5
\end{cases} \]
### Graph Explanation
The given piecewise function \( f(x) \) is graphed on a coordinate plane with the x-axis ranging from -6 to 6 and the y-axis from -6 to 6.
- **Interval \([-1, 0)\):** The function \( f(x) = -x^2 + 1 \) is a downward-opening parabola from \((-1, 0)\). The graph includes \((-1, 0)\) as a filled circle and ends at \( (0, 1) \) as an open circle.
- **Interval \((0, 1):** The function \( f(x) = 4x \) is a linear line continuing up to, but not including, \( x = 1 \), shown with an open circle at \( (1, 4) \).
- **At \(x = 1\):** There is a discrete point \( f(x) = -3 \), marked by a filled circle at \( (1, -3) \).
- **Interval \((1, 3):** The function \( f(x) = -4x + 8 \) creates a line decreasing between these points. The endpoint at \( (3, -4) \) is open.
- **Interval \((3, 5):** A horizontal line at \( f(x) = 4 \) spans this region. Both the start \( (3, 4) \) and end \( (5, 4) \) are open circles.
This visual representation can assist students in determining the continuity of the function over various intervals by showing changes in the function behavior at different points of \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6811bcd-02f8-46fa-b044-8f44b9c4930d%2F8a2cbdb5-f588-4ad1-8e68-97961532f98d%2Fapvn3op_processed.png&w=3840&q=75)
Transcribed Image Text:**Find the intervals on which the function is continuous.**
\[ \text{45)} \quad f(x) = \begin{cases}
-x^2 + 1, & -1 \leq x < 0 \\
4x, & 0 < x < 1 \\
-3, & x = 1 \\
-4x + 8, & 1 < x < 3 \\
4, & 3 < x < 5
\end{cases} \]
### Graph Explanation
The given piecewise function \( f(x) \) is graphed on a coordinate plane with the x-axis ranging from -6 to 6 and the y-axis from -6 to 6.
- **Interval \([-1, 0)\):** The function \( f(x) = -x^2 + 1 \) is a downward-opening parabola from \((-1, 0)\). The graph includes \((-1, 0)\) as a filled circle and ends at \( (0, 1) \) as an open circle.
- **Interval \((0, 1):** The function \( f(x) = 4x \) is a linear line continuing up to, but not including, \( x = 1 \), shown with an open circle at \( (1, 4) \).
- **At \(x = 1\):** There is a discrete point \( f(x) = -3 \), marked by a filled circle at \( (1, -3) \).
- **Interval \((1, 3):** The function \( f(x) = -4x + 8 \) creates a line decreasing between these points. The endpoint at \( (3, -4) \) is open.
- **Interval \((3, 5):** A horizontal line at \( f(x) = 4 \) spans this region. Both the start \( (3, 4) \) and end \( (5, 4) \) are open circles.
This visual representation can assist students in determining the continuity of the function over various intervals by showing changes in the function behavior at different points of \( x \).
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