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