Function Features

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extreme values

Functions are packages containing three sets: domain, codomain, and pairs. The pairs connect the domain values to codomain values and it is this relationship we want to investigate.

For us, the codomain is often $\mathbb {R}$ and so we often ignore this until it becomes important. (There are many reasons for it to become important). Thus, we often think of our real-valued functions as three sets: domain, range, and pairs.

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.

Extreme Function Values

$\blacktriangleright $ Where is the maximum value of the function?

Translation: Which domain value results in the maximum value of the function?

$\blacktriangleright $ Where is the minimum value of the function?

Translation: Which domain value results in the minimum value of the function?

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. The answers are not the points in the graph. The answers are the numbers in the domain and range.

The answers are not the points in the graph.

The answers are the numbers in the domain and range.

Video: Two Types of Questions

[ Click on the arrow to the right to expand the video. ]

The graph displays patterns in its points. These patterns help us analyze the function which has values in the range connected to numbers in the domain.

Maximum

Global Maximum

Let $f$ be a function defined on the domain $D$.

Then $M$ is the global maximum value of $f$ on $D$ if

There exists $d \in D$, such that $f(d) = M$. i.e., $M$ has to be a function value
$f(d) \leq M$ for all $d \in D$. i.e., all function values are less than or equal to $M$.

If there is no such $M$, then $f$ has no maximum value.

Note: The global maximum is often referred to as just the maximum.

Graph vs. Function

The maximum value is visually encoded in the highest point on the graph of the function. The second coordinate is the maximum value and the first coordinate is the domain number where the maximum occurs.

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

$\blacktriangleright $ desmos graph

This browser does not support embedded elements.

The highest point on the graph of $f$ is

$(-5,0)$ $(-2,9)$ $(0,5)$ $(-1,-2)$ $(5,-6)$ $(8,0)$

The maximum value of $f$ is $\answer {9}$, which occurs at $\answer {-2}$.

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

$\blacktriangleright $ desmos graph

This browser does not support embedded elements.

There is no highest point on the graph. Therefore, $g$ has no maximum value.

Minimum

Global Minimum

Let $f$ be a function defined on the domain $D$.

Then $M$ is the global minimum value of $f$ on $D$ if

There exists $d \in D$, such that $f(d) = M$. i.e., $M$ has to be a function value
$f(d) \geq M$ for all $d \in D$. i.e., all function values are greater than or equal to $M$.

If there is no such $M$, then $f$ has no minimum value.

Note: The global minimum is often referred to as just the minimum.

Graph vs. Function

The minimum value is visually encoded in the lowest point on the graph of the function. The second coordinate is the minimum value and the first coordinate is the domain number where the minimum occurs.

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

$\blacktriangleright $ desmos graph

This browser does not support embedded elements.

The lowest point on the graph of $f$ is

$(-5,0)$ $(-2,9)$ $(0,5)$ $(-1,-2)$ $(5,-6)$ $(8,0)$

The minimum value of $f$ is $\answer {-6}$, which occurs at $\answer {5}$.

Below is the graph of $y=H(k)$.

$\blacktriangleright $ desmos graph

This browser does not support embedded elements.

There is no lowest point on the graph. Therefore, $H$ has no minimum value.

Where?

$\blacktriangleright $ A function might not have a global maximum value.

If a function has a global maximum value, it has exactly one global maximum value.

However, this one global maximum value might occur at many domain numbers.

$\blacktriangleright $ A function might not have a global minimum value.

If a function has a global minimum value, it has exactly one global minimum value.

However, this one global minimum value might occur at many domain numbers.

A Weird Example

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

$C$ is a constant function with $[-6,5]$ as its domain.

$C$ has only one function value. That is $2$.

$\blacktriangleright $ $C(x) \leq 2$ for every domain number.

So, $2$ is the global maximum value of $C$, and it occurs at every domain number.

$\blacktriangleright $ $C(x) \geq 2$ for every domain number.

So, $2$ is the global minimum value of $C$, and it occurs at every domain number.

Functions can do weird things.

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

2026-05-23 01:29:54