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       

increasing-function                                

However if

strictly-increasing-function
        ∀ 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                
decreasing-function 

However if

        ∀ x2 > x1 => f(x2) < f(x1); x1, x2 ε D
strictly-decreasing-function 

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 Df = 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

  

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