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