# Behaviors of Functions

### Learning Objectives

- Determine where a function is increasing, decreasing, or constant
- Find local extrema of a function from a graph
- Describe behavior of the toolkit functions

**local maximum**. If a function has more than one, we say it has local maxima. Similarly, a value of the input 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*. A function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of

*local*extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain. For the function below, the local maximum is 16, and it occurs at [latex]x=-2[/latex]. The local minimum is [latex]-16[/latex] and it occurs at [latex]x=2[/latex]. To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. The graph below illustrates these ideas for a local maximum.

### A General Note: Local Minima and Local Maxima

A function [latex]f[/latex] is an**increasing function**on an open interval if [latex]f\left(b\right)>f\left(a\right)[/latex] for any two input values [latex]a[/latex] and [latex]b[/latex] in the given interval where [latex]b>a[/latex]. A function [latex]f[/latex] is a

**decreasing function**on an open interval if [latex]f\left(b\right)<f\left(a\right)[/latex] for any two input values [latex]a[/latex] and [latex]b[/latex] in the given interval where [latex]b>a[/latex]. A function [latex]f[/latex] has a local maximum at [latex]x=b[/latex] if there exists an interval [latex]\left(a,c\right)[/latex] with [latex]a<b<c[/latex] such that, for any [latex]x[/latex] in the interval [latex]\left(a,c\right)[/latex], [latex]f\left(x\right)\le f\left(b\right)[/latex]. Likewise, [latex]f[/latex] has a local minimum at [latex]x=b[/latex] if there exists an interval [latex]\left(a,c\right)[/latex] with [latex]a<b<c[/latex] such that, for any [latex]x[/latex] in the interval [latex]\left(a,c\right)[/latex], [latex]f\left(x\right)\ge f\left(b\right)[/latex].

### Example: Finding Increasing and Decreasing Intervals on a Graph

Given the function [latex]p\left(t\right)[/latex] in the graph below, identify the intervals on which the function appears to be increasing.Answer:
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 (1,3) and the interval [latex]\left(4,\infty \right)[/latex].

#### Analysis of the Solution

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).### Example: Finding Local Maxima and Minima from a Graph

For the function [latex]f[/latex] whose graph is shown below, find all local maxima and minima.Answer:
Observe the graph of [latex]f[/latex]. The graph attains a local maximum at [latex]x=1[/latex] because it is the highest point in an open interval around [latex]x=1[/latex]. The local maximum is the [latex]y[/latex] -coordinate at [latex]x=1[/latex], which is [latex]2[/latex].
The graph attains a local minimum at [latex]\text{ }x=-1\text{ }[/latex] because it is the lowest point in an open interval around [latex]x=-1[/latex]. The local minimum is the *y*-coordinate at [latex]x=-1[/latex], which is [latex]-2[/latex].

## 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] |

## Use a graph to locate the absolute maximum and absolute minimum of a function

There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\text{-}[/latex] coordinates (output) at the highest and lowest points are called the**absolute maximum**and

**absolute minimum**, respectively. To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\left(x\right)={x}^{3}[/latex] is one such function.

### A General Note: Absolute Maxima and Minima

The**absolute maximum**of [latex]f[/latex] at [latex]x=c[/latex] is [latex]f\left(c\right)[/latex] where [latex]f\left(c\right)\ge f\left(x\right)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]. The

**absolute minimum**of [latex]f[/latex] at [latex]x=d[/latex] is [latex]f\left(d\right)[/latex] where [latex]f\left(d\right)\le f\left(x\right)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex].

### Example: Finding Absolute Maxima and Minima from a Graph

For the function [latex]f[/latex] shown below, find all absolute maxima and minima.Answer:
Observe the graph of [latex]f[/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[/latex] and [latex]x=2[/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the *y*-coordinate at [latex]x=-2[/latex] and [latex]x=2[/latex], which is [latex]16[/latex].
The graph attains an absolute minimum at [latex]x=3[/latex], because it is the lowest point on the domain of the function’s graph. The absolute minimum is the *y*-coordinate at [latex]x=3[/latex], which is [latex]-10[/latex].

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