Types of Functions

Constant Function:
Let ‘A’ and ‘B’ be any two non–empty sets, then a function ‘f’ from ‘A’ to ‘B’ is called a constant function if and only if the range of ‘f’ is a singleton.

Algebraic Function:
A function defined by an algebraic expression is called an algebraic function.
e.g. f\left( {\text{x}} \right) = {{\text{x}}^2} + 3{\text{x}} + 6

Polynomial Function:
A function of the form {\text{P}}\left( {\text{x}} \right) = {{\text{a}}_{\text{m}}}{{\text{x}}^{\text{n}}} + {{\text{a}}_{{\text{n}} - 1}}{{\text{x}}^{{\text{n}} - 1}} + \cdots + {{\text{a}}_1}{\text{x}} + {{\text{a}}_0}
where ‘n’ is a positive integer and {{\text{a}}_{\text{n}}},{{\text{a}}_{{\text{n}} - 1}}, \cdots ,{{\text{a}}_1},{{\text{a}}_0} are real numbers is called a polynomial function of degree ‘n’.

Linear Function:
A polynomial function with degree ‘t’ is called a linear function. The most general form of a linear function is
f\left( {\text{x}} \right) = {\text{ax}} + {\text{b}}

Quadratic Function:
A polynomial function with degree ‘2’ is called a quadratic function. The most general form of a quadratic equation is f\left( {\text{x}} \right) = {\text{a}}{{\text{x}}^2} + {\text{bx}} + {\text{c}}

Cubic Function:
A polynomial function with degree ‘3’ is called a cubic function. The most general form of a cubic function is f\left( {\text{x}} \right) = {\text{a}}{{\text{x}}^3} + {\text{b}}{{\text{x}}^2} + {\text{cx}} + {\text{d}}

Identity Function:
Let f:{\text{A}} \to {\text{B}} be a function then ‘f’ is called an identity function if f\left( {\text{x}} \right) = {\text{x,}}\;\forall \;{\text{x}} \in {\text{A}}.

Rational Function:
A function R\left( {\text{x}} \right) defined by R\left( {\text{x}} \right) = \frac{{{\text{P}}\left( {\text{x}} \right)}}{{{\text{Q}}\left( {\text{x}} \right)}}, where both {\text{P}}\left( {\text{x}} \right)and{\text{Q}}\left( {\text{x}} \right) are polynomial functions is called a rational function.

Trigonometric Function:
A function f\left( {\text{x}} \right) = \sin {\text{x}}, f\left( {\text{x}} \right) = \cos {\text{x}} etc., then f\left( {\text{x}} \right) is called a trigonometric function.

Exponential Function:
A function in which the variable appears as an exponent (power) is called an exponential function
e.g. (i) f\left( {\text{x}} \right) = {{\text{a}}^{\text{x}}} (ii) f\left( {\text{x}} \right) = {3^{\text{x}}}.

Logarithmic Function:
A function in which the variable appears as an argument of a logarithm is called a logarithmic function.
e.g. f\left( {\text{x}} \right) = {\log _{\text{a}}}\left( {\text{x}} \right).