Beyond,

the questions asks us

**Quote:**

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

and then we are given 5 different equations. One of these equations will give the same answer for any number X as for (1-x). In other words, if you put 10 into the equation (x) you would get the same answer as if you put -9 (1-10) in.

The simplest way to solve the equation is pick easy numbers to work with, but any numbers you choose will work (by definition they have to because of the line "for ALL x"). I chose 2 for X. If X =2 then (1-x) = (1-2) or -1

then go down the line and plug in 2 for each equation and -1 for each equation. When you find the equation that gives you the same answer for both you're golden! (unless two equations happen to work for a set of numbers, then you pick new numbers and try those 2 again until only one continually works for you).

Let's plug in 2 and -1 then:

1.

f(2) = 1-(2) which gives us -1

f(-1)= 1-(-1) which gives us 2

-1 does not equal 2, so this isn't it

2.

f(2) = 1-(2^2) = 1-4 which gives us -3

f(-1) =1-(-1^2)= 1-1 which gives us 0

-3 does not equal 0, so this isn't it either

etc etc until you get to 4

4.

f(2) = 2^2(1-2)^2

4(-1)^2 = 4(1) = 4

f(-1)= -1^2(1-(-1)^2

1(2)^2 = 1(4) =4

BINGO! both X and 1-X give us the same answer when plugged into equation 4!

now typically this will be your correct answer, but sometimes the same numbers will happen to work for other equations (but only one equation will work for ALL numbers) so you may want to test the other ones even after finding one that works. If you're pressed for time though you'd probably choose D (and be correct!) and move in.