Substitution and equivalence

If f(x2)=x4−6x2+5, which of the following must be true?

1. f(x2)=f(1+x2)
2. f(x2)=f(2−x2)
3. f(x2)=f(3−x2)
4. f(x2)=f(5−x2)
5. f(x2)=f(6−x2)
   

The problem checks your understanding of plugging one expression g(x) into another f(x).

Example

To substitute g(x)=x−1 for x in f(x)=2x2+4x+7, I replace for clarity the dummy input variable by z and put protective parentheses around each of its occurrences. f(z)=2(z)2+4(z)+7f(x−1)=2(x−1)2+4(x−1)+7.

Expanding and simplifying yields f(g(x))=2x2+5.
Observe that the relation z=x−1 can be uniquely solved for x, z=x−1⇔x=z+1,

the replacement g(x) is invertible, and the original function f(z) can be recovered by substituting x=z+1: f(z)=2(z+1)2+5=2z2+4z+7

Back to our problem

The replacement g(x)=x2 isn’t invertible for all real x. Only for x≥0 we have z=x2⇔x=√z. Therefore, we can’t recover f(z) completely. Indeed, even f1(z)={1+4cosz, if z<0z2−6z+5, if z≥0

yields f1(x2)=x4−6x2+5 for all real x. Only non-negative inputs to f are important!
GMAT problems don’t expect you to deal with this kind of intricacies, and it should be safe to take f(z)=z2−6z+5 even for z<0. To recover f(z), I substituted z for x2 in f(x2)=x2⋅x2−6⋅x2+5.

A statement like f(x2)=f(2−x2) has to be translated carefully, f(x2)⇝f(z)⇝f(2−x2),

and means substituting 2−x2 for every occurrence of x2 in f(x2)=x2⋅x2−6⋅x2+5: f(x2)⏞x4−6x2+5=f(2−x2)⏞(2−x2)2−6(2−x2)+5

We don’t have to solve here an equation. The equation is supposed to be true for all input values x, and we only have to check if that’s true. While the test input x=1 does solve the equation, the input x=0 already shows that f(x2)=f(2−x2) is generally false, and we can dismiss answer choice (B).
Similarly, the test input x=0 doesn’t solve any of the equations x4−6x2+5=(1+x2)2−6(1+x2)+5x4−6x2+5=(3−x2)2−6(3−x2)+5x4−6x2+5=(5−x2)2−6(5−x2)+5,

and we can reject answer choices (A), (C), and (D). The only option left is (E). Indeed, expanding f(6−x2) yields (6−x2)2−6(6−x2)+5=36−12x2+x4−36+6x2+5,

which is equivalent to x4−6x2+5.

The upshot

If the answer choices of a GMAT problem contain a variable x (function, equation, inequality), the answer choices are supposed to be true for all values of x. It’s usually cheap and fast to discard wrong answers by plugging in test values.