If function f(x) satisfies f(x) = f(x^2) for all x, which of the follo

If function \(f(x)\) satisfies \(f(x) = f(x^2)\) for all values of \(x\), 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 a particular function \(f(x)\) has the property \(f(x) = f(x^2)\) for all values of \(x\). Note that we don't know the specific form of the function; we only know this one property. For example, for this function, \(f(3)=f(3^2)\) or \(f(3)=f(9)\); similarly, \(f(9)=f(81)\), so \(f(3)=f(9)=f(81)=\ldots\).

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 necessarily true that \(f(2)=f(2)* f(2)\) (this will only be true if \(f(2)=1\) or \(f(2)=0\), but as we don't know the actual function, we cannot say for sure);

B. \(f(16) - f(-2) = 0\): since \(f(-2)=f(4) =f(16)=...\), we have that \(f(16)-f(-2)=f(16)-f(16)=0\), so this option is always true;

C. \(f(-2) + f(4) = 0\): \(f(-2)=f(4)\), but it's not necessarily true that \(f(4) + f(4)=2f(4)=0\) (this will only be true 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 \(f(3)=0\)? Since we don't know the actual function, we cannot say for sure;

E. \(f(0)=0\): Again, as we don't know the actual function, we cannot say for sure.

Alternatively, we can consider a function that satisfies \(f(x) = f(x^2)\) for all values of \(x\). For instance, the function can be \(f(x) = 10\), meaning that for this function, no matter what the input \(x\) is, the output will always be 10. If we interpret this as a graph, we get a horizontal line at \(y = 10\). Now, we can evaluate the options taking \(f(x) = 10\):

A. \(f(4)=f(2)*f(2)\). This option is not true, since \(10 \neq 10*10\).

B. \(f(16) - f(-2) = 0\). This option is true since \(f(16) - f(-2) = 10 - 10 = 0\).

C. \(f(-2) + f(4) = 0\). This option is not true, since \(10 + 10 \neq 0\).

D. \(f(3)=3*f(3)\). This option is not true, since \(10 \neq 3*10\).

E. \(f(0)=0\). This option is not true, since \(f(0) = 10\), not 0.


Answer: B

Hope it's clear.