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the key to solving this problem is that if not given a restriction as in b... the function can go on forever right?

Below is a detailed solution of this problem. Hope it helps.

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

We are told that some function f(x) has following property f(x) = f(x^2) for all values of x. Note that we don't know the actual function, just this one property of it. For example for this function f(3)=f(3^2) --> f(3)=f(9), similarly: f(9)=f(81), so f(3)=f(9)=f(81)=....

Now, the question asks: which of the following MUST be true?

A. f(4)=f(2)*f(2): we know that f(2)=f(4), but it's not necessary f(2)=f(2)*f(2) to be true (it will be true if f(2)=1 or f(2)=0 but as we don't know the actual function we can not say for sure);

B. f(16) - f(-2) = 0: again f(-2)=f(4) =f(16)=... so f(16)-f(-2)=f(16)-f(16)=0 and thus this option is always true;

C. f(-2) + f(4) = 0: f(-2)=f(4), but it's not necessary f(4) + f(4)=2f(4)=0 to be true (it will be true only if f(4)=0, but again we don't know that for sure);

D. f(3)=3*f(3): is 3*f(3)-f(3)=0? is 2*f(3)=0? is f(3)=0? As we don't know the actual function we can not say for sure;

E. f(0)=0: And again as we don't know the actual function we can not say for sure.

When F(x) = F(x²) for all x, then F must be a constant function. That means F(0)=F(1)=F(...) whatever. That is why B must be correct. _________________

........................................................................................ See it big and keep it simple.

It does not provide information about the value of F(x) or F(x^2) .. so we can not perform any operation with F(x) or F(x^2). The leaves us only F(16) - F(-2) = 0 and F(-2)+F(4) = 0 to be examined..

F(x) = F(x^2) => f(-2) = f(4) =f(16) .. so B is the answer.

If for some reason the answer is B, why not A? We are saying F(-2) = F(4) = F(16) and therefore F(16) - F(-2) => F(16) - F(16) =0 must be true. So in the above example we are doing a simple arithmetic of 16 - 16 for F(16) - F(16).

Why not do 4 = 2 * 2 for F(4) = F(2)*F(2)? Can someone please help explain?

If for some reason the answer is B, why not A? We are saying F(-2) = F(4) = F(16) and therefore F(16) - F(-2) => F(16) - F(16) =0 must be true. So in the above example we are doing a simple arithmetic of 16 - 16 for F(16) - F(16).

Why not do 4 = 2 * 2 for F(4) = F(2)*F(2)? Can someone please help explain?

We don't know what the function actually returns; we just know that the function returns the same value for a number and its square.

the key to solving this problem is that if not given a restriction as in b... the function can go on forever right?

Below is a detailed solution of this problem. Hope it helps.

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

We are told that some function f(x) has following property f(x) = f(x^2) for all values of x. Note that we don't know the actual function, just this one property of it. For example for this function f(3)=f(3^2) --> f(3)=f(9), similarly: f(9)=f(81), so f(3)=f(9)=f(81)=....

Now, the question asks: which of the following MUST be true?

A. f(4)=f(2)*f(2): we know that f(2)=f(4), but it's not necessary f(2)=f(2)*f(2) to be true (it will be true if f(2)=1 or f(2)=0 but as we don't know the actual function we can not say for sure);

B. f(16) - f(-2) = 0: again f(-2)=f(4) =f(16)=... so f(16)-f(-2)=f(16)-f(16)=0 and thus this option is always true;

C. f(-2) + f(4) = 0: f(-2)=f(4), but it's not necessary f(4) + f(4)=2f(4)=0 to be true (it will be true only if f(4)=0, but again we don't know that for sure);

D. f(3)=3*f(3): is 3*f(3)-f(3)=0? is 2*f(3)=0? is f(3)=0? As we don't know the actual function we can not say for sure;

E. f(0)=0: And again as we don't know the actual function we can not say for sure.

Answer: B.

Great explanation! I missed out on the initial part and got lost (f'x is the same as f'x2 the same as f'x4) - Great question!

the key to solving this problem is that if not given a restriction as in b... the function can go on forever right?

Below is a detailed solution of this problem. Hope it helps.

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

We are told that some function f(x) has following property f(x) = f(x^2) for all values of x. Note that we don't know the actual function, just this one property of it. For example for this function f(3)=f(3^2) --> f(3)=f(9), similarly: f(9)=f(81), so f(3)=f(9)=f(81)=....

Now, the question asks: which of the following MUST be true?

A. f(4)=f(2)*f(2): we know that f(2)=f(4), but it's not necessary f(2)=f(2)*f(2) to be true (it will be true if f(2)=1 or f(2)=0 but as we don't know the actual function we can not say for sure);

B. f(16) - f(-2) = 0: again f(-2)=f(4) =f(16)=... so f(16)-f(-2)=f(16)-f(16)=0 and thus this option is always true;

C. f(-2) + f(4) = 0: f(-2)=f(4), but it's not necessary f(4) + f(4)=2f(4)=0 to be true (it will be true only if f(4)=0, but again we don't know that for sure);

D. f(3)=3*f(3): is 3*f(3)-f(3)=0? is 2*f(3)=0? is f(3)=0? As we don't know the actual function we can not say for sure;

E. f(0)=0: And again as we don't know the actual function we can not say for sure.

Answer: B.

Hi Bunuel,

thanks for the explanation. I found this question to be very tough and I seem to be the only one not being able to understand this even after your explanation

I could understand why B is the answer and why not C and D. But I am still confused with A and E. I have tried to explain my thought process below -

We know that f(x) = f(x^2) means f(2) = f(4) = f(16) and so on..... A. f(4)=f(2)*f(2) LHS is f(4) which means (16) RHS is f(2)*f(2) means 4*4 means (16) Hence LHS = RHS? As I am writing this, it occurred to me that here we are multiplying to functions {f(2)*f(2)} and we don't really know if multiplying of two functions will actually result in the multiplication of those two numbers/integers? it could result in some other function as well? Am I thinking in the right direction?

Coming to option E - it says f(0)=0 we know that f(x) = f(x^2) if x is 0...its square (infact any exponent) or function of its square should always result in 0. I can't think through what's wrong with E?

Could you please help me understand where I am going wrong. Many thanks for your help.