A function f: R → R satisfies the equation f(x) f(y)− f(xy) = x + y for all x, y ∈ R and f(1) > 0, then
Answer & explanation
Correct answer: option 4
Taking x = y = 1, we get
$f (1) f (1)− f (1) =1+1 ⇒ f(1)^2- f(1) -2 =0$
$⇒ ( f (1)− 2)( f (1)+1) = 0 ⇒ f (1) = 2\,\, (∵ f (1) > 0)$
Taking y = 1 we get
$f (x) f (1)− f (x) = x +1 ⇒ 2 f (x)− f (x) = x +1 ⇒ f (x) = x +1$