I think it is B.

F(16) = F(4) since F(x) = F(x^2) read it the other way
F(-2) = F(4) since F(x) = F(x^2) read it the right way

F(4) - F(4) = 0

Is that correct?

I believe the answer is B.

The function F(x) can be any function so you cannot make any assumptions about the actual nature of the function.

F(x) = F(x^2) so
F(0) = F(0)
F(1) = F(1)
F(2) = F(4); F(-2) = F(4)
F(3) = F(9); F(-3) = F(9)
F(4) = F(16); F(-4) = F(16)
etc, etc.

This is all you really have to work with.

F(-2) = F(4) and F(4) = F(16) so F(-2) = F(16) and thus F(16) - F(-2) = 0.

Answer is B.

good job guys.. OA is B

By the way, since you cannot make any assumptions about the nature of the function in this problem, D and E can be eliminated immediately. They are trap answers:

D) F(3) = 3 * F(3)
E) F(0) = 0

The 3 in D and 0 in E appear to come out of thin air; really what the test writer is hoping is that an unwary person inserts an assumption about the nature of the function. If you know nothing about the function then you cannot possibly have a definitive value (like 3 and 0) in the answer.

In general, there are two distinct parts of these type of function questions. The first is the domain, which are all the values for X in F(X). The second is the actual function itself. The plan of attack should depend on what information is given:

(1) The actual function itself is not given (as in this question) - you must work with the domain, which are the possible values of X.

(2) The function is given - you probably will need to work with both the domain and the function

For situation (1) where the function itself is not given, approach the problem like my explanation on this problem. Find possible values of X in the domain and use the information provided to manipulate F(x) where x is any value in the domain X.

Here's another problem discussed in these forums:

For which of the following functions f is f(x) = f(1-x) for all x?

Pick values of x and plug into both sides of the equation:
f(0) = f(1)
f(1) = f(0)
f(2) = f(-1)

For situation (2) where the function is given, first pick some domain value(s) that are "easy" to use, then plug those values into the function.

The same problem:

For which of the following functions f is f(x) = f(1-x) for all x?

A. f(x) = 1-x
B. f(x) = 1-x^2
C. f(x) = x^2 - (1-x)^2
D. f(x) = (x^2)(1-x)^2
E. f(x) = x / (1-x)

From the discussion above we know f(0) = f(1) so the two values of x that we should be using are 0 and 1. We choose these because they are extremely easy values to plug in given functions in choices A-E;

A. f(0) = 1 - 0 = 1
f(1) = 1 - 1 = 0 ---> f(0) is not equal to f(1). Incorrect.

repeat for B, C, and E. all incorrect

D. f(0) = (0^2)(1-0)^2 = (0)(1) = 0
f(1) = (1^1)(1-1)^2 = (1)(0) = 0 --> so f(0) = f(1). Correct.

Sometimes two values x in the domain X can yield the same result for a given function. This is a property of functions - two values x in the domain X can have the same result. If this is the case, pick a new value of x (hopefully an "easy" value) and retest.

good approach. but here is one more. we need to calc f(1-x) as we are given f(x)

A ) f(1-x) = x not equal
B) f (1-x) = 1 - (1-x) ^2 = 2x -x ^2 not equal
C) f(1-x) = (1-x) ^ 2 - x ^2 = 1-2x not equal
D) f(1-x) = (1-x)^2 (x)^2 equal.