Increasing and Decreasing Function

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

Let’s begin –

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, \(\infty\)).

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.