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