Types of Functions

Types of Functions.Functions and different types of functions. Constant, Identity, Equality, Polynomial, Power, Rational, Irrational, Absolute (Modulus), Signum, Exponential, Integer, Monotonic, Periodic. Composition, Strictly Monotonic, Increasing, Decreasing

Constant Function: 

A function f :X->Y is called a constant function if its range is a singleton set i.e., f(x) = c for all x E X where c is some constant.
constant-function

NOTE : For a function of real variable 


 

Identity Function:

A function f:X->X is called an Identity Function if

identity-function

Equality of Function:

Two functions f:X->Y and g:X->Y are said to be equal iff f(x)=g(x) for all x E X i.e., iff Df = Dg and Rf = Rg.

Polynomial Function:

The function f(x) = a0 + a1x +...+anxn , where a0,...,an are real constants such that an not equal to 0 and "n" is a positive integer, is called a Polynomial Function in "x" with real co-efficients and it is of degree "n".
 

Power Function:

A function f: R->R defined by, f(x)=xa, a E R is called a power function.

Rational Function:

A function defined by the quotient of two polynomial functions is called a rational function. Thus R(x) = P(x)/Q(x), where P(x) and Q(x)[q!=0]are polynomial functions, is called a rational function.

Irrational Function:

A function involving one or more radicals of polynomials is called an irrational function

irrational-function-example

Absolute value Function:

The absolute value function or the numerical value function or the modulus (mod) of a real  number x, denoted by |x|, is defined as 
|x| = x if x>0
or, |x| = -x if x<0 font="">
or, |x| = 0 if x = 0

Properties of Modulus of a real number

moduls-function-property



The function f : R->R defined by f(x) = |x|, is called Modulus Function.
Here, Df = R, Rf = R+

Signum Function:

The Signum of the real number x is defined as

signum-function

The function f:R->R given by f(x) = Sgn(x) is called a Signum function. Here Df and R(f)= {-1, 0 , 1}.

Greatest Integer Function:

For all x E R. Let [x] denote the greatest integer in x not greater integer in x not greater than x.

greatest-integer-function

The function f: R->R given by f(x) = [x] is called greatest integer function or the step function.
Note. Here Df = R and Rf = Z, set of all integers.
 

Exponential Function:

The function f:R->R defined by f(x) = ax, where "a" is a positive real number different from 1, is called an exponential function.
Here Df = R and Rf = R+ - {0}.

Logarithmic Function:

Let f : R+ - 0 ->R be defined by f(x) = logax where a[not equal to 0], and a[not equal to]. Then f is called a logarithmic function.
Here Df = R+ - {0}, Rf = R.

Periodic Function:

A function y=f(x) defined on a domain X is periodic if there exists a constant a>0 such that f(x+a) = f(x-a) =f(x) for all x E X.

Monotonic Function:

Let f: [a,b]->R be a function. The function f is said to be
   i. monotonic increasing on [a, b] if v (x1,x2)E [a,b],monotonic-increasing
              ii.strictly monotonic increasing on [a,b], if v (x1,x2)E[a,b],
strictly-monotonic-increasing
     iii. monotonic decreasing on [a,b] if V x1,x2 E[a,b],x12
monotonic-decreasing
iv. strictly monotonic decreasing on [a,b] if V x1,x2 E[a,b],
 strictly-monotonic-decreasing

Composition of functions:  

Let f : X-> Y be defined by y=f(x)
and g: X-> Z be defined by z= g(y)
then, h: X->Z defined by h(x) = g(f(x)) is called composite  function.

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MathZAG: Types of Functions
Types of Functions
Types of Functions.Functions and different types of functions. Constant, Identity, Equality, Polynomial, Power, Rational, Irrational, Absolute (Modulus), Signum, Exponential, Integer, Monotonic, Periodic. Composition, Strictly Monotonic, Increasing, Decreasing
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