Mathematics - Functions Question with Solution | TestHub

Taking $x=y=1$, we get:

$$f(1)f(1)-f(1)=2$$

$$f^2(1)-f(1)-2=0$$

$$(f(1)-2)(f(1)+1)=0$$

Since $f(1)>0$, we have $f(1)=2$. Taking $y=1$, we get:

$$f(x)\cdot f(1)-f(x)=x+1$$

$$f(x)\cdot 2-f(x)=x+1$$

$$f(x)=x+1$$

Therefore, $f^{-1}(x)=x-1$.

Thus, $f(x)\cdot f^{-1}(x)=(x+1)(x-1)=x^2-1$.

So, (C) is the correct answer.