# JEE Main & Advanced Mathematics Functions Algebra of Functions

## Algebra of Functions

Category : JEE Main & Advanced

(1) Scalar multiplication of a function : $(c\,f)(x)=c\,f(x),$  where $c$ is a scalar. The new function $c\,f(x)$ has the domain ${{X}_{f}}.$

$(f\pm g)(x)=f(x)\pm g(x).$ The new function has the domain $X$.

(3) Multiplication of functions

$(fg)(x)=(g\,f)(x)=f(x)g\,(x).$ The product function has the domain $X$.

(4) Division of functions :

(i) $\left( \frac{f}{g} \right)\,(x)=\frac{f(x)}{g(x)}.$ The new function has the domain $X,$ except for the values of $x$ for which $g\,(x)=0.$

(ii) $\left( \frac{g}{f} \right)\,(x)=\frac{g(x)}{f(x)}.$ The new function has the domain $X,$ except for the values of $x$ for which $f(x)=0.$

(5) Equal functions : Two function $f$ and $g$ are said to be equal functions, if and only if

(i) Domain of $f=$ Domain of $g$

(ii)  Co-domain of $f=$ Co-domain of $g$

(iii) $f(x)=g(x)\,\forall x\in$ their common domain

(6) Real valued function : If $R,$ be the set of real numbers and $A,\,\,B$ are subsets of $R,$ then the function $f:A\to B$ is called a real function or real–valued function.

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