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's use the INPUT-OUTPUT approach.
So, let's use a "nice" value for x.
How about x =
0?
So, we can reword the question as, For which of the following functions is f(
0)=f(1-
0)
In other words, we're looking for a function such that
f(0) = f(1)
A) f(x)=1-x
f(0)=1-0 =
1
f(1)=1-1 =
0
Since f(0) doesn't equal f(1), eliminate A
B) f(x) = 1 - x²
f(0) = 1 - 0² =
1
f(1) = 1 - 1² =
0
Since f(0) doesn't equal f(1), eliminate B
C) f(x) = x² - (1-x)²
f(0) = 0² - (1-0)² =
-1
f(1) = 1² - (1-1)² =
1
Since f(0) doesn't equal f(1), eliminate C
D) f(x) = x²(1-x)²
f(0) = 0^2(1-0)^2 =
0
f(1) = 1^2(1-1)^2 =
0
Since f(0) equals f(1), keep D for now
E) f(x) = x/(1-x)
f(0) = 0/(1-0) =
0
f(1) = 1/(1-1) =
undefined
Since f(0) doesn't equal f(1), eliminate E
Since only
D satisfies the condition that f(x)=f(1-x) when x=0, the correct answer is
D
Here's a similar question:
https://www.beatthegmat.com/number-systems-t270738.html
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
