How to solve this function problem of test prep.

[karan.7045]

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

1)f(X)=1-x
2)f(x)=1-x^2
3)f(x)-x^2-(1-x)^2
4)f(x)=x^2(1-x)^2
5)f(x)=x/(1-x)

[Brent@GMATPrepNow]

For which of the following function 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)

Let's try plugging in an easy value for x. How about x = 0.
So, we can reword the question as, For which of the following functions is f(0)=f(1-0)
In other words, we're looking for a function such that f(0) = f(1)

A) f(x)=1-x
f(0)=1-0 = 1
f(1)=1-1 = 0
Since f(0) doesn't equal f(1), eliminate A

B) f(x) = 1 - x^2
f(0) = 1 - 0^2 = 1
f(1) = 1 - 1^2 = 0
Since f(0) doesn't equal f(1), eliminate B

C) f(x) = x^2 - (1-x)^2
f(0) = 0^2 - (1-0)^2 = -1
f(1) = 1^2 - (1-1)^2 = 1
Since f(0) doesn't equal f(1), eliminate C

D) f(x) = x^2(1-x)^2
f(0) = 0^2(1-0)^2 = 0
f(1) = 1^2(1-1)^2 = 0
Since f(0) equals f(1), keep D for now

E) f(x) = x/(1-x)
f(0) = 0/(1-0) = 0
f(1) = 1/(1-1) = undefined
Since f(0) doesn't equal f(1), eliminate E

Since only D satisfies the condition that f(x)=f(1-x) when x=0, the correct answer is D

Cheers,
Brent

[Matt@VeritasPrep]

You could also approach this from a conceptual perspective. This sort of question bothers a lot of students because it doesn't define a function in the traditional way. On the typical GMAT question, I give you a function - say, f(x) = 3x + 5 - then ask you to do something with that function, such as determine the value of f(9), or whatever.

This question, on the other hand, doesn't define a function; it asks for a function that gives the same output for two different input values. Imagine a question like, "Give a function for which f(9) is equal to f(6)." One answer to that question would be f(x) = x^2 - 15x + 54, because f(6) = 0 and f(9) = 0, so f(6) = f(9).

Our question, of course, is more abstract than my example. It wants a function for which f(x) = f(1-x), a condition that can be tough to even understand at first. But once you get what it's asking for, the question isn't so bad: you just have to find a function for which f(x) = some number and f(1-x) = that same number. (D) is the right answer because f(x) = x^2 * (1-x)^2 and f(1-x) = (1-x)^2 * (1-(1-x))^2, which simplifies to (1-x)^2 * x^2, which is the same thing as f(x), only backwards.

This a great question, by the way: I've had students who ended up scoring 45-48 on the quant section who struggled with it multiple times (i.e. they missed it, got the explanation, then still missed it again a week or two later!), so it's a doozy.