Problem Solving

This problem really frustrated me, even with knowing the answer, I didn't know it worked out.

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

f(x) = 1-x
f(x) = 1 - x^2
f(x) = x^2 - (1-x)^2
f(x) = x^2*(1-x)^2
f(x) = x / (1-x)


The answer is the 4th one. But I still don't see how.

vongdn wrote:This problem really frustrated me, even with knowing the answer, I didn't know it worked out.

For which of the following functions f is f(x) - f(1-x) for all x?

f(x) = 1-x
f(x) = 1 - x^2
f(x) = x^2 - (1-x)^2
f(x) = x^2*(1-x)^2
f(x) = x / (1-x)


The answer is the 4th one. But I still don't see how.

Since the correct answer has to work for all values of x, we can plug in our own value for x.

Plug in 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 - 2^2 = -3.
f(-1) = 1 - (-1)^2 = 0.
Doesn't work.

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

Answer choice D:
f(2) = 2^2 * (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.