JEE Main & Advanced Mathematics Functions Algebra of Functions

(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 \[{{X}_{f}}.\]

(2) Addition/subtraction of functions

\[(f\pm g)(x)=f(x)\pm g(x).\] 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.