JEE Main & Advanced Mathematics Functions Even and Odd Function

Even and Odd Function

Category : JEE Main & Advanced

(1) Even function : If we put \[(-x)\] in place of \[x\] in the given function and if \[f(-x)=f(x)\], \[\forall x\in \] domain then function \[f(x)\] is called even function. e.g. \[f(x)={{e}^{x}}+{{e}^{-x}},\] \[\,f(x)={{x}^{2}},\,\] \[f(x)=x\sin x,\,\]\[\,f(x)=\cos x,\,f(x)={{x}^{2}}\cos x\] all are even functions.

 

 

(2) Odd function : If we put \[(-x)\] in place of \[x\] in the given function and if \[f(-x)=-f(x),\,\,\forall x\in \] domain then \[f(x)\] is called odd function. e.g., \[f(x)={{e}^{x}}-{{e}^{-x}}\], \[f(x)=\sin x,\,f(x)={{x}^{3}}\], \[f(x)=x\cos x,\] \[f(x)={{x}^{2}}\sin x\] all are odd functions.

 

 

Properties of even and odd function

 

 

  • The graph of even function is always symmetric with respect to y-axis. The graph of odd function is always symmetric with respect to origin.

 

  • The product of two even functions is an even function.

 

  • The sum and difference of two even functions is an even function.

 

  • The sum and difference of two odd functions is an odd function.

 

  • The product of two odd functions is an even function.

 

  • The product of an even and an odd function is an odd function. It is not essential that every function is even or odd. It is possible to have some functions which are neither even nor odd function. g.\[f(x)={{x}^{2}}+{{y}^{3}},\,\,f(x)={{\log }_{e}}\,x,\,\,f(x)={{e}^{x}}\].

 

  • The sum of even and odd function is neither even nor odd function.

 

Zero function \[f(x)=0\] is the only function which is even and odd both.



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