**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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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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Question

**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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