If $f(x × f(y)) = \dfrac{f ( x) }y , x, y \in \mathbb R , y \ne 0$, then prove that $f(x) . f (1 /x) = 1$

Question

The following question is taken from the practice set of JEE exam.

If $f(x × f(y)) = \dfrac{f ( x) }y , x, y \in \mathbb R , y \ne 0$, then prove that $f(x) . f (1 /x) = 1$

Putting $x=0$, I get $$f(0)=\frac{f(0)}y$$

From this I conclude $y=1\;\forall x\in \mathbb R$. Is this a correct conclusion?

Therefore, the given equation becomes $$f(x\times f(1))=f(x)$$

From this I conclude $f(1)=1$. Is this correct?

But not able to obtain $f\left(\dfrac1x\right)$.

Answer 1

Since you have observed $f(1) = 1$, we have $$f(1\cdot f(y)) = \frac{f(1)}{y} = \frac{1}{y} \\ f(f(\color{fuchsia}y)) = \frac{1}{\color{fuchsia}y}$$ Then $$f(x)f\left(\frac1x\right) \overset{1/x = f(f(x))}{=} f(x)f(f(\color{fuchsia}{f(x)})) = f(x)\frac{1}{\color{fuchsia}{f(x)}} = 1$$