Bunuel wrote:
snkrhed wrote:
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 [color=#ff0000]f(2)*f(2) means 4*4 means (16)[/color]
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.
Kind regards
f(4)=f(16), not 16 and f(2)*f(2)=f(4)*f(4), not 4*4.