# Increasing and Decreasing Function

Here you will learn what are increasing and decreasing function with examples.

Let’s begin –

## Increasing and Decreasing Function

#### Strictly Increasing Function

A function f(x) is said to be a strictly increasing function on (a, b), if

$$x_1$$ < $$x_2$$ $$\implies$$ $$f(x_1)$$ < $$f(x_2)$$ for all $$x_1$$, $$x_2$$ $$\in$$ (a, b)

Thus, f(x) is strictly increasing on (a, b) if the values of f(x) increase with the increase in the values of x.

Example : Show that the function f(x) = 2x + 3 is strictly increasing function on R.

Solution : Let $$x_1$$ , $$x_2$$ $$\in$$ R and let $$x_1$$ < $$x_2$$. Then,

$$x_1$$ < $$x_2$$ $$\implies$$ $$2x_1$$ < $$2x_2$$

$$\implies$$ $$2x_1$$ + 3 < $$2x_2$$ + 3

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

Thus, $$x_1$$ < $$x_2$$ $$\implies$$ $$f(x_1)$$ < $$f(x_2)$$ for all $$x_1$$, $$x_2$$ $$\in$$ R.

So, f(x) is strictly increasing function on R.

Example : Show that the function f(x) = $$x^2$$ is strictly increasing function on [0, $$\infty$$).

Solution : Let $$x_1$$ , $$x_2$$ $$\in$$ [0, $$\infty$$) and let $$x_1$$ < $$x_2$$. Then,

$$x_1$$ < $$x_2$$ $$\implies$$ $$(x_1)^2$$ < $$x_1x_2$$             [Multiplying both sides by $$x_1$$]             ……(i)

again, $$x_1$$ < $$x_2$$ $$\implies$$ $$x_1x_2$$ < $$(x_2)^2$$             [Multiplying both sides by $$x_2$$]             ……(ii)

from (i) and (ii), we get

$$x_1$$ < $$x_2$$ $$\implies$$ $$(x_1)^2$$ < $$(x_2)^2$$ $$\implies$$ $$f(x_1)$$ < $$f(x_2)$$

Thus, $$x_1$$ < $$x_2$$ $$\implies$$ $$f(x_1)$$ < $$f(x_2)$$ for all $$x_1$$, $$x_2$$ $$\in$$ [0, $$\infty$$).

So, f(x) is strictly increasing function on [0, $$\infty$$).

#### Strictly Decreasing Function

A function f(x) is said to be a strictly decreasing function on (a, b), if

$$x_1$$ < $$x_2$$ $$\implies$$ $$f(x_1)$$ > $$f(x_2)$$ for all $$x_1$$, $$x_2$$ $$\in$$ (a, b)

Thus, f(x) is strictly decreasing on (a, b) if the values of f(x) decrease with the increase in the values of x.

Example : Show that the function f(x) = -3x + 12 is strictly decreasing function on R.

Solution : Let $$x_1$$ , $$x_2$$ $$\in$$ R and let $$x_1$$ < $$x_2$$. Then,

$$x_1$$ < $$x_2$$ $$\implies$$ $$-3x_1$$ < $$-3x_2$$

$$\implies$$ $$-3x_1$$ + 12 < $$-3x_2$$ + 12

$$\implies$$ $$f(x_1)$$ > $$f(x_2)$$

Thus, $$x_1$$ < $$x_2$$ $$\implies$$ $$f(x_1)$$ > $$f(x_2)$$ for all $$x_1$$, $$x_2$$ $$\in$$ R.

So, f(x) is strictly decreasing function on R.

Example : Show that the function f(x) = $$a^x$$, 0 < a < 1 is strictly decreasing function on R.

Solution : Let $$x_1$$ , $$x_2$$ $$\in$$ R and let $$x_1$$ < $$x_2$$. Then,

$$x_1$$ < $$x_2$$

$$\implies$$ $$a^{x_1}$$ < $$a^{x_2}$$ $$\implies$$ $$f(x_1)$$ > $$f(x_2)$$

Thus, $$x_1$$ < $$x_2$$ $$\implies$$ $$f(x_1)$$ > $$f(x_2)$$ for all $$x_1$$, $$x_2$$ $$\in$$ R.

So, f(x) is strictly decreasing function on R.

### Related Questions

Find the interval in which f(x) = $$-x^2 – 2x + 15$$ is increasing or decreasing.

Prove that $$f(\theta)$$ = $${4sin \theta\over 2 + cos\theta} – \theta$$ is an increasing function of $$\theta$$ in $$[0, {\pi\over 2}]$$.

Separate $$[0, {\pi\over 2}]$$ into subintervals in which f(x) = sin 3x is increasing or decreasing.