Mathematics - Functions Question with Solution | TestHub

Given $f(x+y)+f(x-y)=2f(x).f(y)$........(1)

Replacing $x$ by $y$ and $y$ by $x$ in (1), we have

$$F(y+x)+f(y-x)=2f(y).f(x)\Rightarrow f(x+y)+f(y-x)=2f(x).f(y)\dots \dots \left(2\right)$$

From (1) and (2), we have $f(x-y)=f(y-x)$.

Now, putting $y=2x$ we get $f(-x)=f(x)$.

Hence $f(x)$ is an even function.