10. Find the open interval of monotonicity where the following function is increasing, decreasing, or constant. x ≤ 3 x > 3 f(x)= -1, -1, r- 5-2.
10. Find the open interval of monotonicity where the following function is increasing, decreasing, or constant. x ≤ 3 x > 3 f(x)= -1, -1, r- 5-2.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Problem 10: Function Monotonicity**
Find the open interval of monotonicity where the following function is increasing, decreasing, or constant.
\[
f(x) =
\begin{cases}
x - 1, & x \leq 3 \\
5 - x, & x > 3
\end{cases}
\]
**Explanation:**
The function \(f(x)\) is piecewise defined with two different expressions depending on the value of \(x\):
1. For \(x \leq 3\), the function is defined as \(f(x) = x - 1\).
2. For \(x > 3\), the function is defined as \(f(x) = 5 - x\).
To find the intervals of monotonicity, analyze the behavior of \(f(x)\) in these intervals:
- **For \(x \leq 3\):** The function \(f(x) = x - 1\) is a linear function with a slope of 1, which means it is strictly increasing in this interval.
- **For \(x > 3\):** The function \(f(x) = 5 - x\) is a linear function with a slope of -1, which means it is strictly decreasing in this interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5bc85531-a363-494f-87f2-b8404e8545c4%2F43ad270e-6a2a-4c5a-8234-1d5ad5d48078%2Fwqrbl5g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 10: Function Monotonicity**
Find the open interval of monotonicity where the following function is increasing, decreasing, or constant.
\[
f(x) =
\begin{cases}
x - 1, & x \leq 3 \\
5 - x, & x > 3
\end{cases}
\]
**Explanation:**
The function \(f(x)\) is piecewise defined with two different expressions depending on the value of \(x\):
1. For \(x \leq 3\), the function is defined as \(f(x) = x - 1\).
2. For \(x > 3\), the function is defined as \(f(x) = 5 - x\).
To find the intervals of monotonicity, analyze the behavior of \(f(x)\) in these intervals:
- **For \(x \leq 3\):** The function \(f(x) = x - 1\) is a linear function with a slope of 1, which means it is strictly increasing in this interval.
- **For \(x > 3\):** The function \(f(x) = 5 - x\) is a linear function with a slope of -1, which means it is strictly decreasing in this interval.
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