Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
The first step is to figure out what this weird term “idempotent” means in practical terms. The expression f(f(x)), or “f of f of x,” just means that you apply the function twice. So the condition f(f(x)) = f(x) means that when you apply the function twice, you get the same result as when you apply the function once—just plain old f(x). You don’t get any extra change the second time you apply the function.
So, we can rephrase the question. Four of the functions are idempotent—that is, when you apply them twice to any number, you get the same thing out as when you apply the function just once. For which function is that NOT the case?
Now let’s test the answer choices one by one. If you struggle to keep the cases straight, just put in 3 for x and see what you get.
(A) f(x) = x is the identity function. If you give it any number, you get back that same number. For instance, f(3) = 3. So applying this function twice will just keep giving you the same number back. In algebraic terms, f(f(x)) = f(x) = x.
(B) f(x) = –x gives you back the number, multiplied by -1. So, for instance, f(3) = -3. So what is f(f(x))? Try it with 3. f(f(3)) = f(-3) = 3. Notice that it gets you back to the original number 3, but what we want is f(3). In other words, we want -3. So this function is NOT idempotent, and B is the answer.
We should keep going to confirm our thinking.
(C) f(x) = |x| is the absolute value function. If you take the absolute value of some number, then take the absolute value of the result, you get the same thing as when you take the absolute value once.
For example, f(-2) = 2, but f(2) = 2, so f(f(-2)) = 2, the same thing as f(-2).
(D) f(x) = the greatest integer less than or equal to x. This is known as the “floor” function, by the way – it means “round down to the integer below, but not if you’re already at an integer.” For instance, f(3.5) = 3, but f(3) = 3. So after you apply this function once, you don’t get any further change if you apply it a second time.
(E) The same thinking applies to this case as well. Once you round up to the nearest integer above or equal to x, you don’t move any more the second time you apply that function.
The correct answer is B.