Problem Solving

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-x^2
C) f(x)=x^2-(1-x)^2
D) f(x)=x^2(1-x)^2
E) f(x)=x/1-x

Substitute some value for X and this will help is getting the response faster.

Let us take X= 1.

Option A. f(x) = 1-x
f(1) = 0
f(1-1)= f(0) = 1-0 = 1 both are not equal. Skip it

Option B. f(x)=1- x^2
f(1) = 1-1 = 0
f(1-1) = f(0) = 1-0^2 = 1 both not equal. Skip it.

Option C: f(x) = x^2-(1-x)^2
f(1) = 1^2 - (0)^2 = 1-0 = 1
f(0) = 0^2 -(1)^2 = -1 . Not equal skip it.

Option D: f(x) =x^2(1-x)^2
f(1) = 1^2(0)^2 = 0
f(0) = 0^2 (1)^2 = 0 . Both are equal. So this is correct.

Option E: f(x) = x/1-x
f(1) = 1/0 = indefinite
f(0) = 0/1 = 0 . Not equal . So skip it.

Therefore the answer is Option D.

Hope this helps.