Problem Solving

What equation is equal to: f(x)=f(1-x) for all x?

Can someone help explain this?

Hey where are the answers? A,B,C...etc

the answers are:

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

it's asking for which of the following functions give the same result when u use x and x-1. f(x) = f(x-1).

Lets pick D. f(x) = x^2 * (1-x)^2.

Now replace x with x-1 in the above function.

f(x-1) = (1-x)^2 * (1-(1-x))^2 = (1-x)^2 * (x)^2 which is equal to f(x). So pick D.

islandgurl918 wrote:the answers are:

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

Plug in numbers and eliminate. Only for one answer choice will f(x) always equal f(1-x), regardless of the value of x. The other answer choices may work for SOME values of x, but not for every value.
Try x=2, so 1-x = -1. Go down the answer choices and plug in 2, then plug in -1, and eliminate every answer choice for which the two values are not equal.
For example, answer choice A:
x=2 f(2) = 1-2 = -1
1-x=-1 f(-1 = 1-(-1)=2
Since th results of the function for the two values is not equal, A cannot be the right answer choice. Eliminate it.
D is the only answer choice for which f(2) is equal to f(1): (-1)^2*2^2 is equal 2^2*(-1)^2. This alone is not enough to prove that D is always true, but since the other answer choices are eliminated, D MUST be the right answer by POE.

Thanks Geva Stern ; I was also taken back by this uestion when i tried to solve it algeberacialy. could you suggest how to hit this in alebric way please.

GMATMadeEasy wrote:Thanks Geva Stern ; I was also taken back by this uestion when i tried to solve it algeberacialy. could you suggest how to hit this in alebric way please.

Sorry, but no.
I believe trying algebra here is the surefire way to spend 5 minutes on the question with a high chance of getting it wrong at the end by falling for one of the trap answer choice. If you focus on D, you can see that it's going be the same for x and 1-x - the two just change places, not values. But you need something to say "look at D!" in order to do that - there's just too much clutter when your first look at the answer choices. The way I outlined above is exactly the way I would've gone about it - I'd use plugging in to easily eliminate the wrong answer choices, so that what remains must be the right one.

Rahul@gurome wrote:if we put (1 - x) in place of x, ....

Let me add some remarks on Rahul´s (correct) solution.

The proper way of defining a function is through a triade ("triple data"), that is, its domain, its codomain and the rule that associates each x (of the domain) to a unique value y (in the codomain) that we usually denote by f(x).

Without taking too much attention to the previous paragraph (although it is nice to remember that some important Mathematicians believe the concept of a function is the most important one of ALL Mathematics...), when we are staring at the "rule" y = f(x) the students are often "under the pressure" of looking into x as something "special", not only a parameter that could have ANY other letter representing it.

What I mentioned above is not irrelevant, on the contrary. You should look at (say) f(x) = 2x^3 as you would look at (say) f(something) = 2 (something) ^3, where "something" is the parameter you would like to choose.

Why this is nice? Because now it is trivial to accept that if you want to calculate f(1-x) , you could (and SHOULD) imagine that the "something" is the 1-x , therefore it´s really just a matter of substitution, therefore f(1-x) = 2 (1-x)^3, in the example discussed above.

In the problem originally presented, you must (and I believe you SHOULD) apply this idea for each alternative, till you discover that the right answer will present the very same final expression for the "x" substitution and the "1-x" substitution (in the rule given by that alternative choice).

I´ve decided to explain this in detail because I guess to many people believe students should avoid concepts a bit more refined like this one, during their preps and/or during the exam itself. I respect that, but I disagree. You (test taker) should NOT take more than 1.5 minutes to find the answer doing it "kosher", and if you try and you get confused or take too long, that only means that you should study and understand all this with care, taking time, for the purpose of doing this sort of reasoning easily and quick during the "real deal", the GMAT test itself!

In other words: (in my opinion) the test taker should DEVELOP his/her quant MATURITY, because the GMAT will detect it one way or the other. Although I believe there ARE students who do well looking for shortcuts or take-aways -- I would bet the majority don´t, otherwise the GMAT average scores would be much greater, because the "easier path" is the natural choice for the majority of test takers -- I have no doubts that students who KNOW what they are doing WILL have much greater possibilities of getting very high marks and... they will have the PLEASURE to know what they are doing, and will have the self-confidence that it seems (to me) impossible to get without this knowledge. Think about it!

Best Regards,
Fabio.

Thanks all for your helpful replies!