Function Behavior

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relative change

Functions are packages containing three sets: domain, codomain (or range), and pairs. The pairs connect the domain numbers to range values and it is this relationship we want to investigate. We would like to know how the range values are affected by the domain values.

We are interested in the function values - the range values. But they are controlled by the domain values.
We ask questions about the function values, but the answers are in the domain.

Function Behavior

$\blacktriangleright $ Where is the function increasing?

Translation: On which set of domain values do the function values increase?

$\blacktriangleright $ Where is the function decreasing?

Translation: On which set of domain values do the function values decrease?

A graph provides a global view of all of the pairs, which reveals many of the patterns, features, and characteristics we seek.

However, we must separate the graph from the function. The graph helps us answer the questions, but the graph doesn’t hold the answers. The answers are in the domain and range.

Increasing

Increasing

Let $f$ be a function defined on the domain $D$.
Let $S \subset D$ be any subset of $D$.

$f$ is increasing on $S$ provided $f$ possesses this property:

For every pair $a, b \in S$, when $a \leq b$ then $f(a) \leq f(b)$.

Increasing means the domain and range change the same: both become greater or both become lesser.

Visually, increasing looks like the points of the graph of $f$ are going uphill to the right.

Visually, increasing looks like the points of the graph of $f$ are going downhill to the left.

Below is the graph of $y=f(x)$.

$\blacktriangleright $ desmos graph

This browser does not support embedded elements.

Select all of the subsets upon which $f$ is increasing.

$(-5,0)$ $(-2,0)$ $(-2,0) \cup (1,5)$ $(1,5)$ $(1,5) \cup (5,8)$ $(5,8)$

The definition of increasing never mentions breaks in the graph.

In the example above, there is a gap in $(-2,0) \cup (1,5)$. However, for the domain numbers in this set, the associated function values follow the definition of increasing.

The set $(1,8)$ is another story.

On $(1,8)$, we have

$4 \in (1,8)$ and $6 \in (1,8)$ with $4 \leq 6$
however, $f(4) > f(6)$

The function is not increasing on the set $(1,8)$, even though it is increasing on $(1,5)$ and $[5,8)$ separately.

Decreasing

Decreasing

Let $f$ be a function defined on the domain $D$.
Let $S \subset D$ be any subset of $D$.

$f$ is decreasing on $S$ provided $f$ possesses this property:

For every pair $a, b \in S$, when $a \leq b$ then $f(a) \geq f(b)$.

Decreasing means the domain and range change oppositely: one becomes greater and the other becomes lesser.

Visually, decreasing looks like the points of the graph of $f$ are going downhill to the right.

Visually, decreasing looks like the points of the graph of $f$ are going uphill to the left.

Below is the graph of $y=w(t)$.

$\blacktriangleright $ desmos graph

This browser does not support embedded elements.

Select all of the subsets upon which $f$ is decreasing.

$(-7,-2)$ $(-2,6)$ $[-2,6)$ $(6,8)$ $[6,8]$ $(-7,6)$ $(-2,8)$

If a function is increasing on its whole domain, then we just say the function is increasing.
If a function is decreasing on its whole domain, then we just say the function is decreasing.

This story is a bit misleading. Our story involved $\leq $. Technically, the equal sign allows the function to stay steady or constant.

An increasing function could increase in value or stay the same. The points could go uphill to the right or stay at the same height.

For this reason, we have the word strictly.

Strictly Increasing

Let $f$ be a function defined on the domain $D$.
Let $S \subset D$ be any subset of $D$.

$f$ is strictly increasing on $S$ provided $f$ possesses this property:

For every pair $a, b \in S$, when $a < b$ then $f(a) < f(b)$.

Strictly Decreasing

Let $f$ be a function defined on the domain $D$.
Let $S \subset D$ be any subset of $D$.

$f$ is strictly decreasing on $S$ provided $f$ possesses this property:

For every pair $a, b \in S$, when $a < b$ then $f(a) > f(b)$.

Strictly increasing or decreasing functions cannot keep the same value. They must change as the domain changes.

Function Behavior

Increasing and decreasing together make up the function’s behavior.

Function behavior is probably the most important function characteristic to investigate.

One of the main goals of Calculus is optimization.

When you model a relationship between two measurements, you are often interesting in maximum and minimum values. The method for identifying these rests on function behavior.

The general connection is

At maximums, functions switch from increasing to decreasing
At minimums, functions switch from idecreasing to increasing

Thus, identifying extreme values and where they occur in the domain is at the top of our to-do list.

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more examples can be found by following this link
More Examples of Function Graphs

2026-05-23 01:08:33