Types of Functions

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.

NOTE : For a function of real variable 

 

Identity Function:

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

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 D_f = D_g and R_f = R_g.

_{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)=x^a, 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}

_{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}

_{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}}

^{_{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.}}

_{^{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],}}

              _{^{ii.strictly monotonic increasing on [a,b], if v (x1}},_^x2)E[a,b],

     iii. monotonic decreasing on [a,b] if V x₁,x₂ E[a,b],x₁2

iv. strictly monotonic decreasing on [a,b] if V x₁,x₂ E[a,b],

 

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.