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.

Hope it's clear.
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You cannot rewrite each answer as a function of x. If I tell you, for example, that f(2) = 2, there are infinitely many possible functions for f(x). It might be that f(x) = x, or that f(x) = 2, or that f(x) = x^2 - 2, or that f(x) = 17x - 32, or that f(x) = 3^x - 7, to give a few examples. When you are told about a specific value of a function -- for example, if you're told that f(5) = 10 -- that only gives you a single point on the graph of y = f(x) (all you know is that the graph contains the point (5,10)). It gives you no information at all about the rest of the graph, and therefore gives very little information about the definition of the function. So in the attached question, you will not be able to determine what f(x) is from any of the answer choices. Instead you need to use the property given -- that f(x) = f(x^2) -- to see which answer choice can be proven to be true. If f(x) = f(x^2), then f(-2) = f(4), and applying this again, f(4) = f(16), which makes B the correct answer. We still don't know what f(x) is, however.
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Rather than analyzing each answer, I would like to point out how the correct answer should look like.
From the equation \(f(x) = f(x^2)\) we can get a chain of equalities between the values of the function \(f\) at different points.
So, we will be able to deduce different equalities of the type \(f(a)=f(b)\), but there is no way to find explicit values of the function in any specific point.
The correct answer should be of this form, or its equivalent \(f(a)-f(b)=0\).
Only answer B is of this type.

For any specific value of \(x\), except \(0\) and \(1\), we can start an infinite chain of equalities. For example, start with \(x=2\):

\(f(2)=f(4)=f((-2)^2)=f(-2)=f(4^2)=f(16)=f((-4)^2)=f(-4)=f(16^2)=f(256)=f((-16)^2)=f(-16)=...\).
We can see that for a given \(x\), the function \(f\) will have the same value at all the points \(x, x^2,x^4,x^8,..., -x,-x^2,-x^4,-x^8,...\)

For \(0\), we just get \(f(0)=f(0^2)\), while for \(x=1\), we have \(f(1)=f((-1)^2)=f(-1)\).
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[quote="tonebeeze"]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. \(f(0) = 0\)

A- f(4) = f(2)

Now is it always true that f(2)f(2) = f(2) OR f(2) = 1
It might be or might not be true.

B - f(16) = f(-2)
f(16) = f(4) = f(2) = f(-2)
This must be true

C. f(-2) = -f(4)
f(-2) = f(4). It might or might not be equal to negative of it.

D. f(3) = 3f(3) Again this might not be true always

E. f(0) = 0 We don't know what will be the y value of the function as expressed in option C, D and E.

Hence the answer is B.
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