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

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

by sbhawal » Fri Nov 29, 2013 12:03 am
I came across this problem while solving the GMAT Prep

Question: For which of the following functions f(x) = f (1-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)

Answer: D

Other than substituting values for x and checking each, what is the shorter alternative/method of arriving at the answer? It took me more than 5 mins to check all the options. I cannot use the same method during the exam. Pls help!!
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by theCodeToGMAT » Fri Nov 29, 2013 12:21 am
let x = 1

{A}
f(x) = 0
f(1-x) = 1 - (1-1) = 1
NO

{B}
f(x) = 1 - (1)^2 = 0
f(1-x) = 1 - (1-x)^2 = 1
NO

{C}
f(x) = 1 - 0 = 1
f(1-x) = (1-1)^2 - (1 - 1 + 1)^2 = 0 - 1 = -1
NO

{D}
f(x) = 1(1-1)^2 = 0
f(1-x) = (1-1)^2 (1-1+1)^2 = 0
YES

{E}
f(x) = 1/0 =ND
f(1-x) = 0
NO

Answer [spoiler]{D}[/spoiler]
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by Brent@GMATPrepNow » Fri Nov 29, 2013 7:22 am
sbhawal wrote: [/b] For which of the following functions f(x) = f (1-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)

Other than substituting values for x and checking each, what is the shorter alternative/method of arriving at the answer? It took me more than 5 mins to check all the options. I cannot use the same method during the exam. Pls help!!
When posting questions, please use the spoiler function to hide the correct answer. This will allow others to attempt the question without seeing the final answer.

What numbers did you plug in? If you choose something like x = 7 or x = -5.4, it will take a while to evaluate each expression. However, if you choose a nice value for x (e.g., x = 0 or x = 1), then your calculations shouldn't take long at all.

Here's my solution:

Let's try plugging in an easy 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 - x^2
f(0) = 1 - 0^2 = 1
f(1) = 1 - 1^2 = 0
Since f(0) doesn't equal f(1), eliminate B

C) f(x) = x^2 - (1-x)^2
f(0) = 0^2 - (1-0)^2 = -1
f(1) = 1^2 - (1-1)^2 = 1
Since f(0) doesn't equal f(1), eliminate C

D) f(x) = x^2(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

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Mathsbuddy » Fri Nov 29, 2013 8:47 am
Note: f(1-x) = -f(x-1) (unless squared)

If f(x) is our starting graph, then -f(x-1) is a vertical reflection that is translated to the right one unit.

What shape can be reflected vertically and moved right without changing? Only f(x) = 0

As this is not on the list, it must be a square function:
Only (D) f(x) = x^2( 1-x)^2 is totally square.

Let's check it:

f(x) = x^2( 1-x)^2
Replace x with (1-x):
f(x-1) = (1-x)^2(1-(1-x))^2 = x^2( 1-x)^2

CORRECT!
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