Question about functions

Question

Given that f(0) is not equal to 0, and that x and y are real numbers. If 2 f(x) f(y) = f(x+y) + f(x-y), express f(2x) in terms of f(x).

Am I missing some info? The answer is

2([f(x)]^2 + 1)

(I only have this part of the question so I'm not sure if I can find out the answer with this info only)

Answer 1

You got it right!

Let y = 0

2f(x)f(0) = f(x) + f(x) = 2f(x) => f(0) = 1

Set y=0 in the original equation.

2f(x)f(0) = f(x+0)+f(x-0) = f(x)+f(x) = 2f(x)

Divide by 2f(x) to get f(0)=1

Then do what you did to start: set y=x, 2f(x)f(x) = f(2x) + f(0)

f(2x) = 2f²(x) - f(0) = 2f²(x) - f(0) = 2f²(x) - 1

Not the answer you gave, but I believe it is correct. If you get a different explanation in class, please let me know. Thank you.