Function f(x)

[melguy]

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)
(B) f(16) - f(-2) = 0
(C) f(-2) + f(4) = 0
(D) f(3) = 3f(3)
(E) f(0) = 0

If x=4:
f(4) = f(4^2)
f(4) = f(16)
So f(4) = f(16).

If x= -2:
f(-2) = f((-2)^2)
So f(-2) = f(4).

But in option 2 f(16) should be written as f(256) instead of f(4) [f(4) would give correct answer] as the function squares all the value. So why did we not square 16 and instead used the square root of 16?

Please help with the problem. Thanks!

[srijoyray]

As I see it,

You can square 16 and write it as f(256). But you have to realize that you can keep increasing the function f(-2) i.e. f(-2) = f(4) = f(16) = f(256).

So (B) becomes f(256) - f(256) = 0. No matter how we solve, only option B holds true.

[ganeshrkamath]

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)
(B) f(16) - f(-2) = 0
(C) f(-2) + f(4) = 0
(D) f(3) = 3f(3)
(E) f(0) = 0

If x=4:
f(4) = f(4^2)
f(4) = f(16)
So f(4) = f(16).

If x= -2:
f(-2) = f((-2)^2)
So f(-2) = f(4).

But in option 2 f(16) should be written as f(256) instead of f(4) [f(4) would give correct answer] as the function squares all the value. So why did we not square 16 and instead used the square root of 16?

Please help with the problem. Thanks!

f(x) = f(x^2)_____________________(1)

Option A: f(4) = f(16)
Also f(x^2) = f(x)_____________________(2)
f(4) = f(2) = f(-2)
But we don't know whether f(2)f(2) = f(4)
Eliminate.

Option B: f(16) - f(-2)
= f(256) - f(4) (no information can be obtained)
But, from (2), f(16) is also equal to f(4).
So f(16) - f(-2) = f(4) - f(4) = 0

Choose B

Cheers

[GMATGuruNY]

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)
(B) f(16) - f(-2) = 0
(C) f(-2) + f(4) = 0
(D) f(3) = 3f(3)
(E) f(0) = 0

Start by examining f(-2) and f(4), since each is included in two of the five answer choices.

f(-2):
Since f(x) = f(x²), we get:
f(-2) = f ( (-2)² )
f(-2) = f(4).

f(4):
Since f(x) = f(x²), we get:
f(4) = f(4²)
f(4) = f(16).

Since f(-2) = f(4), and f(4) = f(16), f(-2) = f(16) -- proving that answer choice B must be true:
f(16) - f(-2) = 0
f(16) = f(-2).

The correct answer is B.