Mathematics - Functions Question with Solution | TestHub

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

Given $f(0)=0$

(i) put $x=0,y=\frac{1}{x}$ we get

$$f(x)+f(-x)=2f(0)f(x)$$

$$\Rightarrow f(x)+f(-x)=0\Rightarrow f(x)=-f(-x)..\dots \left(ii\right)$$

(i) put x = 1, y = 1

$f(2)+f(0)=2[f(1){]}^2$

$f(2)=2[f(1){]}^2\dots (A)$

(i ) put x = -1, y = -2