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Important Functions

1.   Absolute value function (modulus function):
 = magnitude of or the positive value of .
The expression  can be further split as follows:
 =     

Continuity
The graph of  is continuous (i.e. no break in the curve) but has a corner at origin as shown.

Domain & Range
The Domain of the function  is  and Range is

Greatest integer function (unit step function)

 = the greatest integer less than or equal to .

It can also be simplified as:
          if          where n is an integer.


Continuity

The graph of f (x) is discontinuous (i.e. break in the curve) at integral value of x.

Domain & Range

The Domain of the function  is  and Range is

Signum Function

 =     


This can also be written as

 =



 

Continuity
The graph of  is continuous for all values of  except at  where there is a break in the curve.

Domain & Range
The Domain of the function is  and Range is set

Exponential Function

           where          or             









This function is the inverse of logarithmic function i.e. it can be obtained by interchanging
 by  in

As observed from the graph, if  then  increases as  increases.

If, then decreases as increases.

Continuity

 





The graph of
 is continuous (i.e. no break in the curve) everywhere.

Domain & Range

The Domain of the function  is  and Range is.

Trigonometric Functions

If a function f (x) is symmetrical about origin i.e. f (x) =- f (-x) for all values of x, then the function y = f (x) is called an odd function.


Period

Period of   and  is

Period of  is

Continuity
The graph of   and  is continuous (i.e. no break in the curve) everywhere.
The graph of  is discontinuous (i.e. break in the curve) at x = (2n + 1)

Domain & Range
The Domain of   is  and Range is set [-A, A].
The Domain of  is   and Range is.

 





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