Problem 45 - Precalculus with Limits

Question

In Exercises 39-46, determine the intervals over which the function is increasing, decreasing, or constant. \( f(x) = \left\\{ \begin{array}{ll} x+3, & \mbox{ \) x \le 0 \(} \\ 3, & \mbox{ \) 0 < x \le 2 \(} \\ 2x+1, & \mbox{ \) x > 2 \(} \end{array} \right.\)

Step-by-Step Solution

Answer

The function \(f(x)\) is increasing on the intervals \(-\infty \le x \le 0\) and \(2 < x < \infty\), and constant on the interval \(0 < x \le 2\).

Step 1: Check first interval

In the first interval \(x \le 0\), the function is defined as \(f(x) = x+3\). This a linear function with a positive slope (\(m = 1\)). Therefore, within this interval, the function is increasing. This is because a positive slope signifies that as \(x\) becomes larger, so too does \(f(x)\).

Step 2: Check second interval

In the second interval \(0 < x \le 2\), the function is defined as a constant, \(f(x) = 3\). Since it's a constant function, it doesn't increase or decrease, thus it's constant in this interval.

Step 3: Check third interval

In the third interval \(x > 2\), the function is defined as \(f(x) = 2x + 1\). This is a linear function with positive slope (\(m = 2\)). Therefore, within this interval, similar to step 1, the function is increasing.

Key Concepts

Increasing and Decreasing Intervals

Understanding the increasing and decreasing intervals of a function is fundamental in analyzing its behavior. Intuitively, if the output of a function (typically represented as f(x)) grows as the input x increases, the function is said to be increasing on that interval. Conversely, if the value of f(x) decreases when x gets larger, the function is decreasing in that segment.

In our piecewise function f(x), the first segment is defined by f(x) = x + 3 for x ≤ 0. Since for every step to the right on the x-axis (which means increasing x), the f(x) value rises, this segment of the function is increasing. Similarly, in the third segment where the function is defined as f(x) = 2x + 1 for x > 2, an increase in x leads to a higher output of f(x), making it also an increasing interval. There is no interval where the function decreases, so the decreasing interval for this function is non-existent.

Constant Function

A constant function is a type of function where the value of f(x) does not vary no matter how x changes; it remains the same throughout the domain. That is, for any input x, a constant function returns the same output. This property makes it stand out on a graph as a horizontal line.

In the context of our problem, during the second interval where 0 < x ≤ 2, the function takes the form f(x) = 3. This flat line on the graph indicates no variation or movement up or down as x changes; thus, it is a true representation of a constant function. It's important to highlight that a constant function's rate of change is zero, hence it is neither increasing nor decreasing.

Linear Functions

A linear function is perhaps one of the simplest and most identifiable types of functions in algebra. Its most defining characteristic is that it creates a straight line when graphed. The general form of a linear function is f(x) = mx + b, where m is the slope and b is the y-intercept. The slope indicates the steepness and direction of the line; a positive slope means the line ascends as x increases, while a negative slope indicates it descends.

In our piecewise function, both the first section (f(x) = x + 3 for x ≤ 0) and third section (f(x) = 2x + 1 for x > 2) are linear functions with positive slopes. This is evident from the coefficient of x (which is 1 in the first interval, and 2 in the third), ensuring that both segments of the function will plot as ascending lines on a graph.