Problem Solving

Hey guys I tried searching around. I know this is a fairly straight forward problem but I have spent way too much time on it.

Thanks guys!

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)

[spoiler]
The Answer is D[/spoiler]

D is correct

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

hence f(x)=f(1-X)

basically we need to substitute 1-x for x in the eqn and check the value if it is same as of fx then its correct

Why don't you insert in each of these options (1-x) instead of x and then you'll see.
It's really easy.
For example, let's take B
f(x)=1-x^2
f(1-x) = 1-(1-x)^2=1-1+2x-x^2= 2x-x^2
Hence, f(1-x)doesn't equal f(x)

Following this rule - you'll see D is right

Thank you guys for your quick replies. You have helped me out a lot.

Is there a faster way to get the answer instead of just plug 1-x into each of the answer choices?


Thank you!

virtuoso902 wrote:Is there a faster way to get the answer instead of just plug 1-x into each of the answer choices?


Thank you!

we can also use some simple values like x=1
and then check each option

ofcourse, substitute values, take x=2 then f(x)=2 and f(1-x)=-1

check for all given conditions and only option D satisfies it :

f(2)=x^2(1-x)^2 = 4(1-2)^2 = 4
f(-1)= (-1)^2 * (1+1)^2 = 4

Thank you guys for your very helpful replies.

I've gathered that this is probably the fastest way to do this problem. Let me know if I am wrong.

I will take 2 substitute it for X and just plugging it into all of the answer choices.

Is this the fastest way? Should i start with answer choice C and move down or just go down the list?

Thank you!