Increasing or Decreasing Function

 

The function f is said to be an increasing function in its domain D if

∀ x₂ > x₁ => f(x₂) > f(x₁); x₁, x₂ ε D 

However if

 ∀ x₂ > x₁ => f(x₂) > f(x₁), x₁, x₂ ε D                

The function 'f' is said to be strictly increasing

The function 'f' is said to be decreasing function in its domain D if

∀ x₂ > x₁ => f(x₂) < f(x₁); x₁, x₂ ε D

 

However if

∀ x₂ > x₁ => f(x₂) < f(x₁); x₁, x₂ ε D

Then it said to be strictly decreasing.

Strictly increasing and decreasing functions are also called Monotonic Function.

Illustration:

Is y = 2x + 3 increasing/decreasing function.

Solution:

Since, ∀ x ε R, y ε R

Therefore D_f = R

Let x₂ > x₁; x₁, x₂ ε R

        =>     2x₂ > 2x₁

        =>     2x₂ + 3 > 2x₁ + 3

        =>     f(x₂) > f(x₁)

        =>     y = f(x) = 2x + 3 is strictly increasing function.

Examples:

1. Are the following function increasing/decreasing?

        (a)    y = x3 + 8                 (b)    y = -2x + 4
Ans.  (a) Decreasing

        (b) Strictly decreasing 

You can participate in the online tests conducted by askIITians and also give answers in AQAD (A Question  A Day) just by visiting the website askIITians.com. This will give you a number of benefits in the online live classroom courses targeting IIT JEE 2010 and 2011. Read online Study material for IIT JEE and AIEEE preparation absolutely free at askIITians.com.

To read more, Buy study materials of Set Relations and Functions comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.