A function f : R → R satisfies the equation f (x + y) = f (x) f (y) for all x, y in R and f (x) ≠ 0 for any x in R. Let the function be differentiable at x = 0 and f ' (0) = 2. Show that f '(x) = 2 f (x) for all x in R. Hence, determine f (x).
We have, f(x + y) = f (x). f (y) for all x, y ∈ R.
∴ f (0) = f (0). f (0) ⇒ f (0) {f (0) – 1} = 0
