Question

'f ' is a real valued function not identically zero, satisfying f(x + y) + f(x - y) = 2f(x). f(y) ∀ x, y ∈ R. f(x) is definitely

• A. Odd
• B. Even
• C. Neither even nor odd
• D. None of these

Answer 1

Answer: (B)

Even

Answer 2

Putting x = 0, y = 0 in the functional equation we get 2f(0) = 2f²(0) ⇒ f(0) = 0, 1. But if f(0) is equal to zero then f(x) = 0 ∀ x ∈ R. Thus f(0) = 1. Now putting x = 0 in the functional equation we get, f(y) + f(-y) = 2f(0). f(y)

⇒ f(y) = f(-y). Thus f(x) is even.