Problem Solving

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 a version where we can 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

Cheers,
Brent

Hi nsuen,

Functions are really just equations for graphs. As an example, these two equations mean the same thing:

Y = 2X + 1
f(X) = 2X + 1

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

GMAT assassins aren't born, they're made,
Rich

asherkunal wrote:For which of the following function of 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)

Answer: D

Hi asherkunal,

The equality f(x)=f(1-x) implies that if you swap 'x' with '1-x', you get the same result.

Observing closely, we see that only option D qualifies this characteristic. Option D: f(x) = x^2(1-x)^2

We see that for RHS: x^2*(1-x)^2, if 'x' and '(1-x)' exchange their place, you get the same result, thus it is the correct answer.

The correct answer: D

Hope this helps!

Relevant book: Manhattan Review GMAT Math Essentials Guide

-Jay
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asherkunal wrote:For which of the following function of 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)

Since we are not given any restrictions on the value of x, let's let x = 1. Thus, we are determining for which of the following functions is f(1) = f(1-1), i.e., f(1) = f(0). Next, we can test each answer choice using our value x = 1.

A. f(x) = 1 - x

f(1) = 1 - 1 = 0

f(0) = 1 - 0 = 1

Since 0 does not equal 1, A is not correct.

B. f(x) = 1 - x^2

f(1) = 1 - 1^2 = 1 - 1 = 0

f(0) = 1 - 0^2 = 1 - 0 = 1

Since 0 does not equal 1, B is not correct.

C. f(x) = x^2 - (1 - x)^2

f(1) = 1^2 - (1 - 1)^2 = 1 - 0 = 1

f(0) = 0^2 - (1 - 0)^2 = 0 - 1 = -1

Since 1 does not equal -1, C is not correct.

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

f(1) = 1^2*(1 - 1)^2 = 1(0)= 0

f(0) = 0^2*(1 - 0)^2 = 0(2) = 0

Since 0 equals 0, D is correct.

Alternate Solution:

Let's test each answer choice using x and 1 - x.

A. f(x) = 1 - x

f(x) = 1 - x

f(1 - x) = 1 - (1 - x) = x

Since 1 - x does not equal x, A is not correct.

B. f(x) = 1 - x^2

f(x) = 1 - x^2

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

Since 1 - x^2 does not equal 2x - x^2, B is not correct.

C. f(x) = x^2 - (1 - x)^2

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

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

Since 2x - 1 does not equal 1 - 2x, C is not correct.

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

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

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

Since x^2*(1 - x)^2 equals (1 - x)^2*x^2, D is correct.

Answer: D