Mathematics - Functions Question with Solution | TestHub

Given $\mathrm{f}(\mathrm{x}+\mathrm{y})+\mathrm{f}(\mathrm{x}-\mathrm{y})=2\mathrm{f}(\mathrm{x}).\mathrm{f}(\mathrm{y})$...........(1) Replacing x by y and y by x in (1), we have $F(y+x)+f(y-x)=2f(y)\cdot f(x)\Rightarrow f(x+y)+f(y-x)=2f(x)\cdot f(y)$........(2) From (1) and (2), we have $\mathrm{f}(\mathrm{x}-\mathrm{y})=\mathrm{f}(\mathrm{y}-\mathrm{x})$.

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

Hence $\mathrm{f}(\mathrm{x})$ is an even function.