Types of Functions
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TYPES OF FUNCTIONS
1. RATIONAL FUNCTIONS
This function is defines as the ratio of two polynomials
y = (Anxn + An-1xn-1 + ………..+ A1x + a0 )/(Bnxn + Bn-1xn-1 + ………..+ B1x + a0)
Example: y = (x2+ 4)/(x-4)
2. IRRATIONAL FUNCTIONS
The functions involving radicals are called irrational functions.
Example: y= (3x2+4x+2)1/2
3. ALGEBRAIC FUNCTIONS
A function which consists of a finite number of terms involving powers, And roots of the independent variable x and the four fundamental mathematical operations (+, -, * , /) is called an algebraic function
Example: y = 3x2+4x+2, y = (2x2+2)1/2
From the above definition it is clear that both rational and irrational functions are algebraic functions.
4. TRANSCENDENTAL FUNCTIONS
Any function which is not an algebraic function is called a transcendental function. Trigonometric, inverse trigonometric, exponential and logarithmic functions are examples of transcendental functions.
5. EXPLICIT AND IMPLICIT FUNCTIONS
A variable y is said to be an explicit function of another variable x when y is expressed directly in terms of x.
Example: y = ax2+bx+c
When the relation between two variables x and y is expressed in the form f(x,y)= 0, i.e. an equation which is expressed both in terms of x and y mixed up on either or both the sides, then the function is called an implicit function.
Example: ax2 + by2 + 2hxy + 2gx + 2fy + c = 0
6. ODD FUNCTION
A function f: x èY said to be an odd function if
f(-x) = - f(x) for all x E X
Example: If f(x) = sin x. Then f(-x) = sin (-x) = -sin x = -f(x) for all x E R
7. EVEN FUNCTION
A function f: x èY is said to be an even function if
f(-x) = f(x) for are x E X
Example: If f(x) = cos x Then Cos(-x) = cos(x)
8. IDENTITY FUNCTIONS
If the function f : x èY is defined by f (x) = x for all x E X, then f is said to be the identity function on X.
9. PERIODIC FUNCTIONS
A function f : x èy is said to be periodic with period p > 0 if f(x+p) = f(x) for every x E X.
If f is periodic with period p. then f (x+2p) = f((a+p) + p)
= f(x+p)
= f(x) {by definition of periodic function}
==> 2p is also a period of the function
f (x+3p) = f(x+2p+p) = f (x + 2p)= f(x) { from above result}
So 3p is also a period. And so on.
So if p is a period, then 2p, 3p, 4p,. . . . . are also periods of f
Again consider f(x-p) = f((x-p) + p) {applying f(x) = f(x+p) for all x. Here x is replaced by x-p}
= f(x) for all x
So if p is a period, then so is -p.
Similarly -p, -2p, -3p, . . . . . are also periods of f.
In general, if f is periodic with period p, then all integral multiples of p are also period. The smallest positive value of p satisfying f(x+p) = f(x) for all x is called the fundamental period or simply the period of f.
By this definition, period of sin x and cos x is 2p, period of tan x is p.
NOTE: The graph of any periodic function is repetitive. The length of the interval from one cycle to the next is the period. Thus, graphs of all the trigonometric functions are repetitive.