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If function f(x) satisfies f(x)=f(x^2) for all x, which of the following must be true?

A. f(4)=f(2)*f(2)

f(2) = f(2^2) - From the equation f(x) = f(x^2)
So, f(2) = f(4).
Now, let us check.
Is f(4) = f(2)*f(2) ?
Is f(4) = f(4)*f(4) ? (Since, f(2) = f(4))
Since, the original function is unknown, the equation f(4)=f(4)*f(4) is not necessarily true. If f(4) = 0 or 1 then the equation holds true. If f(4) is any other number then the equation doesn't hold true.

B. f(16)-f(-2)=0

f(2) = f(-2^2) - From the equation f(x) = f(x^2)
f(2) = f(4)
f(4) = f(4^2) - From the equation f(x) = f(x^2)
f(4) = f(16)
Now, let us check.
Is f(16)-f(-2) = 0 ?
Is f(16)-f(4) = 0 ? (Since, f(-2) = f(4))
Is f(16) - f(16) = 0 ? (Since f(4) = f(16))
Hell yeah! The value of f(16) is equal to f(16), irrespective of the original function. So, Bis the correct answer choice.

C. f(-2)+f(4)=0

f(-2) = f(-2^2) - From the equation f(x) = f(x^2)
f(-2) = f(4)
Now, let us check.
Is f(-2)+f(4)=0 ?
Is f(4) + f(4) = 0 (Since f(-2) = f(4))
Is 2*f(4) = 0
If f(4) = 0, then the value of f(-2)+f(4) is equal to 0 else it is not. So, f(-2)+f(4) is not necessarily equal to 0.

D. f(3)=3f(3)

Is f(3) = 3*f(3)
Is f(3) - 3 f(3) = 0 ?
Is -2*f(3) = 0 ?
If f(3) = 0, then the value of f(3)=3f(3) else it is not. So, f(3) = 3*f(3) is not necessarily true.

E. f(0)=0

Since, we don't know the original function, f(0) is not necessarily equal to 0.

Let me know if you need any further help.
p.s: Assumption in red.

das.ashmita wrote:If function f(x) satisfies f(x)=f(x2) for all x, which of the following must be true?
A. f(4)=f(2)*f(2)
B. f(16)−f(−2)=0
C. f(−2)+f(4)=0
D. f(3)=3f(3)
E. f(0)=0

OA: B

the only way to solve this question is to be familiar with periodic property of the functions
unless you are aiming math PhD and giving math subject test/gre you won't remember this property and apply it here

f(x)=f(x^2) should be recognized as the function's body and formula at the same time. That is on the continuous basis you should get f(a)=f(a^2)=f((a)^2)=f(((a^2)^2)^2), so on and on
the smallest (main) period here is x^2. So we count f(2)=f(16) since f(2)=f(2^2)=f((2)^2)^2=f(16)

the other answer choices will give different arguments of the function and we have no periodic property then. This question is the way-way beyond simple GMAT math tricks as it requires looking into complexities of function properties.