Behaviors of Functions

Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant

As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways.

We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. The graph below shows examples of increasing and decreasing intervals on a function.

While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the output where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a local maximum. If a function has more than one, we say it has local maxima. Similarly, a value of the output where a function changes from decreasing to increasing as the input variable increases is called a local minimum. The plural form is “local minima.” Together, local maxima and minima are called local extrema, or local extreme values, of the function. (The singular form is “extremum.”) Often, the term local is replaced by the term relative. In this text, we will use the term local.

Given the function [latex]p\left(t\right)[/latex] in the graph below, identify the intervals on which the function appears to be increasing.

Show Solution

We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from [latex]t=1[/latex] to [latex]t=3[/latex] and from [latex]t=4[/latex] on.In interval notation, we would say the function appears to be increasing on the interval [latex](1,3)[/latex] and the interval [latex]\left(4,\infty \right)[/latex].
[latex]\\[/latex]
Analysis of the Solution
[latex]\\[/latex]Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at [latex]t=1[/latex] , [latex]t=3[/latex] , and [latex]t=4[/latex] . These points are the local extrema (two minima and a maximum).

Analyzing the Toolkit Functions for Increasing or Decreasing Intervals

We will now return to our toolkit functions and discuss their graphical behavior in the table below.

Function	Increasing/Decreasing	Example
Constant Function[latex]f\left(x\right)={c}[/latex]	Neither increasing nor decreasing
Identity Function[latex]f\left(x\right)={x}[/latex]	Increasing
Quadratic Function[latex]f\left(x\right)={x}^{2}[/latex]	Increasing on [latex]\left(0,\infty\right)[/latex]Decreasing on [latex]\left(-\infty,0\right)[/latex] Minimum at [latex]x=0[/latex]
Cubic Function[latex]f\left(x\right)={x}^{3}[/latex]	Increasing
Reciprocal[latex]f\left(x\right)=\frac{1}{x}[/latex]	Decreasing [latex]\left(-\infty,0\right)\cup\left(0,\infty\right)[/latex]
Reciprocal Squared[latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex]	Increasing on [latex]\left(-\infty,0\right)[/latex]Decreasing on [latex]\left(0,\infty\right)[/latex]
Cube Root[latex]f\left(x\right)=\sqrt[3]{x}[/latex]	Increasing
Square Root[latex]f\left(x\right)=\sqrt{x}[/latex]	Increasing on [latex]\left(0,\infty\right)[/latex]
Absolute Value[latex]f\left(x\right)=|x|[/latex]	Increasing on [latex]\left(0,\infty\right)[/latex]Decreasing on [latex]\left(-\infty,0\right)[/latex] Minimum at [latex]x=0[/latex]