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by Brent@GMATPrepNow » Fri Dec 05, 2014 5:36 pm
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
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by ceilidh.erickson » Tue Dec 09, 2014 10:09 am
Brent's approach is by far the best one to use here. However, you could also solve this algebraically: plug (1 - x) in for x in each of the answer choices, and see which one simplifies back to the same function:

A) f(x) = 1 - x
f(1 - x) = 1 - (1 - x)
f(1 - x) = x
Not the same function

B) f(x) = 1 - x²
f(1 - x) = 1 - (1 - x)²
f(1 - x) = 1 - (1 - 2x + x²)
f(1 - x) = 2x - x²
Not the same function

C) f(x) = x² - (1 - x)²
f(1 - x) = (1 - x)² - (1 - (1 - x))²
f(1 - x) = 1 - 2x + x² - x²
f(1 - x) = 1 - 2x
Not the same function

D) f(x) = x²(1 - x)²
f(1 - x) = (1 - x)²(1 - (1 - x))²
f(1 - x) = (1 - 2x + x²)(x)²
f(1 - x) = (1 - x)²(x²)
This is the same function!

E) f(x) = x/(1-x)
f(1 - x) = (1 - x)/(1 - (1 - x))
f(1 - x) = (1 - x)/x
Not the same function

So, we can prove algebraically that the answer is D.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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by MartyMurray » Sat Dec 13, 2014 10:05 am
Brent@GMATPrepNow wrote:
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
I too used this approach and found one thing I want to add.

I was using multiple inputs, changing them for each answer choice, and then finally I realized how much simpler and better it is to keep using the same pair of numbers in all the answer choices, until possibly they generated twin answers in more than one answer choice, at which point it would be time to try other pairs of inputs.
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by Brent@GMATPrepNow » Sat Dec 13, 2014 10:24 am
Exactly. One input is all you need UNLESS you find that that input satisfies more than one answer choice.

Cheers,
Brent
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