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theories of formulas

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theories of formulas

by lukaswelker » Tue Apr 01, 2014 1:23 am
For which of the following functions is f(x) = f(1-x) for all x

f(x)= 1-x
f(x)= 1-x2
f(x)= x2-(1-x)2
f(x)= x2(1-x)2
f(x)= x/(1-x)

notes: the 2 are exponents

Can anybody tell by which formula can this problem be resolved?

Many thanks
Lukas
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by GMATGuruNY » Tue Apr 01, 2014 2:08 am
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)
Let x=2.
Then f(x) = f(2) and f(1-x) = f(1-2) = f(-1).
The question becomes:

For which of the following functions does f(2) = f(-1)?

Answer choice A:
f(2) = 1-2 = -1.
f(-1) = 1-(-1) = 2.
Doesn't work.

Answer choice B:
f(2) = 1 - 2² = -3.
f(-1) = 1 - (-1)² = 0.
Doesn't work.

Answer choice C:
f(2) = 2² - (1-2)² = 4 - 1 = 3.
f(-1) = (-1)² - [1-(-1)]² = 1-4 = -3.
Doesn't work.

Answer choice D:
f(2) = 2² * (1-2)² = 4 * 1 = 4.
f(-1) = (-1)² * [1-(-1)]² = 1 * 4 = 4.
Success!

Answer choice E:
f(2) = 2/(1-2) = -2.
f(-1) = (-1)/[(1-(-1)] = -1/2.
Doesn't work.

The correct answer is D.
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by ceilidh.erickson » Tue Apr 01, 2014 8:34 am
You could solve this problem algebraically. For each of the answer choices, substitute (1 - x) for every instance of (x) to find f(1 - x). Then simplify and see if the two are equal:

a. f(x)= 1-x
f(1 - x) = 1 - (1 - x), so f(1 - x) = x --> not equal

b. f(x)= 1-x²
f(1 - x)= 1 - (1 - x)², so f(1 - x) = 2x - x² --> not equal

c. f(x)= x² - (1 - x)² , so f(x) = 2x - 1
f(1 - x)= x² - (1 - (1 - x))², so f(1 - x) = 0 --> not equal

d. f(x)= x²(1-x)² , so f(x) = x²(1 - 2x + x²)
f(1 - x)= (1 - x)²(1 - (1 - x))², so f(1 - x) = (1 - 2x + x²)(x²) --> equal!

e. f(x)= x/(1-x)
f(1 - x)= x/(1 - (1 - x)) , so f(1 - x) = 1 --> not equal


However, this is a MUCH more time-consuming approach. Picking numbers as Mitch did is a much better way to solve. Personally, I'd always pick 1 or 0, as those often are easier to calculate, but it doesn't make much difference.
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