Problem Solving

This one has to be fairly simple, but I'm just not certain how to solve it.

For which of the following functions of 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

The official answer is d.

The answer to this problem would probably pop out more obviously if it were expressed the way the official test screen would have it.

The trick to this question is to do the following:

1) Replace all "x" with the term, "1 - x"
2) Simplify your new equation to compare it with the equations in the answer choices
3) Since this is a "which of the following" type question, begin with answer choice E.

Example: Let's look at D (since we know it's correct)
D: f(x) = x^2(1-x)^2

Therefore, f(1-x) = (1-x)^2 (1-(1-x))^2
= (1-x)^2 (x)^2
= f(x)

This does not work out for all other answer choices (I checked).

You're the best, Beeparoo. That's really clear--thank you!

An easier way would be to just plug in 2 random numbers...

beny wrote:An easier way would be to just plug in 2 random numbers...

Thanks for the reminder, beny. I'm so used to working out equations that I really forget to pick numbers or even use the backsolving approach. The brute force method was instilled in me after so many years of engineering school... It's like trying to break a bad habit.

I wish there was a patch I could wear... A little desperate? Yes. But GMAT is just so damn intensive.

It's when I plug in numbers that I go wrong again...

For example, when I plug 3 into the function.

f(3)=f (1-3)
f(3)=f(-2)

right?

Then, when I plug it into a)

f(3)=1-3 =-2 That works.

When I plug the same numbers into d), the correct answer, it doesn't work (unless I'm going wrong somewhere).

f(3)=3^2(1-3)^2
f(3)=9(-2)^2
f(3)=9(4)
f(3)=36

Where am I going wrong?

Thanks again...

a.)
f(3) = 1-3 = -2
f(-2) = 1-(-2) = 3
NOT EQUAL

d.)
f(3) = 3^2*(-2)^2 = 36
f(-2) = (-2)^2*3^2 = 36
EQUAL

Got it... Thanks much.