Mathematics - Functions Question with Solution | TestHub

Let $f\left(x+\frac{1}{y}\right)+f\left(x-\frac{1}{y}\right)\dots ..(i)=2f(x).f\left(\frac{1}{y}\right)\forall x,y\in R$

Given $f(0)=0$ ..........(i)

put $x=0,y=\frac{1}{x}$

we get $f(x)+f(-x)=2f(0)f(x)$ $\Rightarrow f(x)+f(-x)=0\Rightarrow f(x)=-f(-x)\dots \dots (ii)$

(i) ⇒ put $x=1,y=1$ $f(2)+f(0)=2[f(1){]}^2$ $f(2)=2[f(1){]}^2\dots \dots .\mathrm{A})$

(i) ⇒ put $x=-1,y=-2$ $f(-2)=2f(-1)f(-1)\Rightarrow -f(2)=2(1{)}^2\dots \dots B$

For $A\&Bf(2)=-f(2)\Rightarrow f(2)=0\Rightarrow f(1)=0$

$\therefore f(1)=f(2)=0$

$\therefore f(1)+f(2)=0$