Deciphering a GMAT Math Question: Solving f(x)=f(1-x) for All x

[Xori]

OK, so I'm taking this GMAT practice test online, and I got stuck on this question. The software tells me the correct answer, but not the explanation. What I'm really stuck on is the wording of the question - it's been 10 years since my last calculus class, and even though it's an algebra question the whole "f(x)" thing screws me up. Can someone please interpret this for me and tell me what the hell the question is asking?

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For which of the following functions f is f(x) = f(1-x) for all x?
A) f(x) = 1 - x
B) f(x) = 1 - x²
C) f(x) = x² - (1 - x)²
D) f(x) = x²(1 - x)²
E) f(x) = x/(1 - x)

 

[dmatador]

f(x) is just the function f evaluated at x. This question wants to know which of the listed functions is such that f(x) gives the same output as f(x - 1).

 

[Mentallic]

Well if f(x)=1-x for example, then f(2x)=1-2x, f(ax+b)=1-(ax+b) etc. so all you have to do is find which function f(x) is equivalent to f(1-x) by substituing 1-x for x everywhere in the equation and then simplifying.

 

[Xori]

Well if f(x)=1-x for example, then f(2x)=1-2x, f(ax+b)=1-(ax+b) etc. so all you have to do is find which function f(x) is equivalent to f(1-x) by substituing 1-x for x everywhere in the equation and then simplifying.

So I should substitute (1-x) for (x) in every of the 5 choices and see which still equals (1-x)?

 

[gb7nash]

It still sounds like you're misunderstanding. Think of x has a place holder. f(x) just means put an x where x is (so you're really not doing anything). For example, for A), f(x) = 1-x. f(1-x) means to put a 1-x where x is. So just plug in 1-x everytime you see x in the equation. f(5) means to put a 5 where x is. etc..

So looking at part A) and setting up f(x) = f(1-x), we have:

1-x = ...?

 

[Xori]

It still sounds like you're misunderstanding. Think of x has a place holder. f(x) just means put an x where x is (so you're really not doing anything). For example, for A), f(x) = 1-x. f(1-x) means to put a 1-x where x is. So just plug in 1-x everytime you see x in the equation. f(5) means to put a 5 where x is. etc..

So looking at part A) and setting up f(x) = f(1-x), we have:

1-x = ...?

Ah, I think I get it.

So to evaluate choice (A), I would do the following(?):
f(x) = 1 - x <-- set up provided by (A)
f(1 - x) = 1 - (1 - x) <--- plugging in (1-x) for x
f(1 - x) = x <--- restating

So now I have the two following statements:
f(x) = 1 - x
f(1 - x) = x

If I set the the two equal to each other, I would have:
1 - x = x
...which is obviously not true for all x, so A is incorrect. So then I repeat for B - D.

Am I doing it right?

 

[gb7nash]

Ah, I think I get it.

So to evaluate choice (A), I would do the following(?):
f(x) = 1 - x <-- set up provided by (A)
f(1 - x) = 1 - (1 - x) <--- plugging in (1-x) for x
f(1 - x) = x <--- restating

So now I have the two following statements:
f(x) = 1 - x
f(1 - x) = x

If I set the the two equal to each other, I would have:
1 - x = x
...which is obviously not true for all x, so A is incorrect. So then I repeat for B - D.

Am I doing it right?

Exactly! Sounds like you're getting the idea. Now do the same thing for the rest of them.

 

[Xori]

Exactly! Sounds like you're getting the idea. Now do the same thing for the rest of them.

Ok, just did it for all of em and got the right answer (D)!

Although I see that picking small numbers and plugging would be a better strategy on test day, but wanted to understand the notation first :P

Thanks so much for your help!

 

[AlephZero]

Although I see that picking small numbers and plugging would be a better strategy on test day, but wanted to understand the notation first :P

The best strategy for this particular problem is to see that if you replace x by 1-x, then you also replace 1-x by 1-(1-x), which is the same as replacing 1-x by x

So the right answer will be unchanged if you swap 1-x and x in the definition of f(x), and the only one where that is true is D.

But that's the difference between getting a good score by following a standard method (which is what you learned to do in this thread), and scoring 100% with time to spare - so don't lose any sleep over it!