tonebeeze wrote:
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. \(f(0) = 0\)
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 \(f(1)=f((-1)^2)=f(-1)\).
_________________
PhD in Applied Mathematics
Love GMAT Quant questions and running.