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GMAT Practice exam 2 - Functions

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by GMATGuruNY » Sat Jun 04, 2016 3:02 am
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 x=2.
Then f(x) = f(2) and f(1-x) = f(1-2) = f(-1).
The question becomes:

For which of the following functions does f(2) = f(-1)?

Answer choice A:
f(2) = 1-2 = -1.
f(-1) = 1-(-1) = 2.
Doesn't work.

Answer choice B:
f(2) = 1 - 2² = -3.
f(-1) = 1 - (-1)² = 0.
Doesn't work.

Answer choice C:
f(2) = 2² - (1-2)² = 4 - 1 = 3.
f(-1) = (-1)² - [1-(-1)]² = 1-4 = -3.
Doesn't work.

Answer choice D:
f(2) = 2² * (1-2)² = 4 * 1 = 4.
f(-1) = (-1)² * [1-(-1)]² = 1 * 4 = 4.
Success!

Answer choice E:
f(2) = 2/(1-2) = -2.
f(-1) = (-1)/[(1-(-1)] = -1/2.
Doesn't work.

The correct answer is D.
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by Brent@GMATPrepNow » Sat Jun 04, 2016 8:45 am
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)
Mitch's solution is perfect. Here's another version where we test the values using 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

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by Brent@GMATPrepNow » Sat Jun 04, 2016 8:46 am
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)
Here's another way to show that D is the correct answer:

Given: f(x) = x²(1-x)²

Now compare f(x) and f(1-x)

f(x) = x²(1-x)² = x²(1-x)²

f(1-x) = (1-x)²[1-(1-x)]²
= (1-x)² x²

Since x²(1-x)² = (1-x)² x², we can be certain that f(x) = f(1-x) for all values of x

Cheers,
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by Brent@GMATPrepNow » Sat Jun 04, 2016 8:47 am
Mitch and I have demonstrated the two methods (Algebraic and Input-Output) for solving a question type I call Variables in the Answer Choices.

If you'd like more information on these approaches, we have some videos:
- Variables in the Answer Choices - https://www.gmatprepnow.com/module/gmat- ... /video/933
- Tips for the Algebraic Approach - https://www.gmatprepnow.com/module/gmat- ... /video/934
- Tips for the Input-Output Approach - https://www.gmatprepnow.com/module/gmat- ... /video/935

Here are some more questions to practice with:
- https://www.beatthegmat.com/what-mistake ... 76293.html
- https://www.beatthegmat.com/car-dealer-s ... 74136.html
- https://www.beatthegmat.com/ps-rate-times-t276107.html
- https://www.beatthegmat.com/to-find-the- ... 73338.html
- https://www.beatthegmat.com/y-dollars-fr ... 73031.html
- https://www.beatthegmat.com/meters-in-x- ... 73854.html
- https://www.beatthegmat.com/a-better-exp ... 79396.html


Cheers,
Brent
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by [email protected] » Sat Jun 04, 2016 9:05 am
Hi lucas211,

Functions are really just equations for graphs. Function questions can often be solved by TESTing VALUES or doing algebra.

Here, we're asked which of these functions has the same RESULT when you plug in (X) and (1-X). This is a perfect situation for TESTing VALUES.

Since we're dealing with functions/graphs, we can TEST any value for X that we want. Let's keep things super-easy and use X = 0.

So, we now need to plug in 0 and 1-0 = 1 into each equation and track the results...

Answer A:
X = 0 ===> 1
X = 1 ===> 0
Not the same result

Answer B:
X = 0 ===> 1
X = 1 ===> 0
Not the same result

Answer C:
X = 0 ===> 0 - 1 = -1
X = 1 ===> 1 - 0 = 1
Not the same result

Answer D:
X = 0 ===> 0(1) = 0
X = 1 ===> 1(0) = 0
SAME RESULT

Answer E:
X = 0 ===> 0/1
X = 1 ===> 1/0
Not the same result

Final Answer: D

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by Matt@VeritasPrep » Tue Jun 07, 2016 10:51 pm
One other clue here is that 1 - (1 - x) = x, for any value of x. Looking at the answers, we want one that returns x and (1 - x) terms when we plug in x OR (1 - x), so C and D are the best candidates. We only need to run f(1 - x) in the function to see if it gives the same thing as the answer too, so that saves a lot of time.
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