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

Cheers,
Brent

Hi nsuen,

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

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

nsuen wrote: ↑

Sun Jun 12, 2016 8:49 pm

Hi-

Q: For which of the following function f is f(x)= f(1-x) for all x? The correct answer is D

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)

I still have no clue how to do this question. Thanks!

Solution:

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 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