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  4. If f is any function, then (1/2)[f(x)+f(-x)] is always:

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Q. If $ f $ is any function, then $ \frac{1}{2}[f(x)+f(-x)] $ is always:

Bihar CECEBihar CECE 2005

Solution:

Let $ g(x)=\frac{1}{2}[f(x)+f(-x)] $
$ \Rightarrow $ $ g(-x)=\frac{1}{2}[f(-x)+f(x)] $
$ \Rightarrow $ $ g(x)=g(-x) $
$ \therefore $ It is an even function.
Note: In any function, if we replace $ x $ by the function does not change the sign, that is an $ -x, $ even function.