Engineering Mathematics - Increasing, Decreasing and Constant Functions

Previously you learned about functions, graph of functions. In this lesson, you will learn about some function types such as increasing functions, decreasing functions and constant functions. These concepts are explained with examples and graphs of the specific functions where ever necessary.

Increasing, Decreasing and Constant Functions

Functions are increasing, decreasing and constant when you plot the graph of the function in a coordinate system. Let’s define the meaning of these functions.

Increasing function

A function $f(x)$ is increasing in an interval for any $x_1$ and $x_2$

if $x_1 < x_2$
implies $f(x_1) < f(x_2)$

Example:

Let $f(x) = x^2 + 1$ be a function. Find all the values for the function to plot the graph.

x	f(x)	(x, f(x))
-1	2	(-1, 2)
0	1	(0, 1)
1	2	(1, 2)
2	5	(2, 5)

The graph of the function will look like the following.

Figure 1 - Increasing Functions — Figure 1 – Increasing Functions

In the above graph, the function is increasing between the interval of (0, 2).

The value of $x_1$ is 0 and $x_2$ is 3,

The value of $f(x_1)$ is 1 and $f(x_2)$ is 5.

Therefore,

$x1 < x2$ implies $f(x_1) < f(x_2)$ is true and it is an increasing function.

Decreasing Function

A function $f(x)$ is decreasing in an interval for any $x_1$ and $x_2$

if $x_1 < x_2$
implies $f(x_1) > f(x_2)$

Example:

Consider a function $f(x) = x^2$ . The function is a parabola. Let’s draw the graph of this function in a Cartesian plane or co-ordinate system.

Before plotting the graph, you need to find points for the graph of the function. A table of points is given below.

x	f(x)	(x, f(x)
-2	4	(-2, 4)
-1	1	(-1, 1)
0	0	(0, 0)
1	1	(1, 1)
2	4	(2, 4)

The graph of the parabola is given below.

Figure 2 - Decreasing Functions — Figure 2- Decreasing Functions

In the above graph, the function is decreasing between the interval ( -2, 0).

The value of $x_1$ is -2 and the value of $x_2$ is 0.

The value of $f(x_1)$ is 4 and the value of $f(x_2)$ is 0.

Then

$x_1 < x_2$ is true and $f(x_1) > f(x_2)$ is also true. Hence, the function is a decreasing function between the interval $(-2, 0)$ .

Constant Function

The function is a constant function in an interval for some $x_1$ and $x_2$

if $x_1 < x_2$
implies $f(x_1) = f(x_2)$

This is simplest form of graph of a function and such a function is always a straight line on the coordinate system.

Let $f(x) = 3$ be the constant function. It means for any value of $x$ in the domain, the value of $f(x)$ is 3.

The graph of constant function is given below.

Figure 3 - Constant Function — Figure 3 – Constant Function

In the above graph of the constant function.

The value of $x_1$ is 1 and the value of $x_2$ is 2.

The value of $f(x_1)$ is 3 and the value of $x_2$ is also 3.

Therefore,

$x_1 < x_2$ and $f(x_1) = f(x_2)$ implies that the function is a constant function.