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 Constant Function if and only if range of ‘f’ is a singleton.
Algebraic Function:
The function defined by algebraic expression are called 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 number 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 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 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 cubic function. The most general form of 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 on 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 function is called, rational function.
Trigonometric Function:
A functionf\left( {\text{x}} \right) = \sin {\text{x}}, f\left( {\text{x}} \right) = \cos {\text{x}}etc, then f\left( {\text{x}} \right)is called trigonometric function.
Exponential Function:
A function in which the variable appears as 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 logarithmic is called logarithmic function.
e.g. f\left(  {\text{x}} \right) = {\log _{\text{a}}}\left( {\text{x}} \right).

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