# 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
$f\left( {\text{x}} \right) = {\text{ax}} + {\text{b}}$

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. $f\left( {\text{x}} \right) = {\log _{\text{a}}}\left( {\text{x}} \right)$.