For which of the following functions f is \(f(x) = f(1-x)\) for all \(x?\)

[VJesus12]

For which of the following functions f is \(f(x) = f(1-x)\) for all \(x?\)

A. \(f(x)=1−x\)

B. \(f(x)=1−x^2\)

C. \(f(x)=x^2−(1−x)^2\)

D. \(f(x)=x^2(1−x)^2\)

E. \(f(x)=\dfrac{x}{1−x}\)

[spoiler]OA=D[/spoiler]

Source: GMAT Prep

[Jay@ManhattanReview]

For which of the following functions f is \(f(x) = f(1-x)\) for all \(x?\)

A. \(f(x)=1−x\)

B. \(f(x)=1−x^2\)

C. \(f(x)=x^2−(1−x)^2\)

D. \(f(x)=x^2(1−x)^2\)

E. \(f(x)=\dfrac{x}{1−x}\)

[spoiler]OA=D[/spoiler]

Source: GMAT Prep

Basically the question asks if we swap \(x\) with \((1-x)\), do we get the same expression?

So only option D works. \(x^2(1−x)^2 = (1−x)^2\cdot x^2\)

Correct answer: D

Hope this helps!

-Jay
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[Scott@TargetTestPrep]

For which of the following functions f is \(f(x) = f(1-x)\) for all \(x?\)

A. \(f(x)=1−x\)

B. \(f(x)=1−x^2\)

C. \(f(x)=x^2−(1−x)^2\)

D. \(f(x)=x^2(1−x)^2\)

E. \(f(x)=\dfrac{x}{1−x}\)

[spoiler]OA=D[/spoiler]

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 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

[Brent@GMATPrepNow]

For which of the following functions f is \(f(x) = f(1-x)\) for all \(x?\)

A. \(f(x)=1−x\)

B. \(f(x)=1−x^2\)

C. \(f(x)=x^2−(1−x)^2\)

D. \(f(x)=x^2(1−x)^2\)

E. \(f(x)=\dfrac{x}{1−x}\)

[spoiler]OA=D[/spoiler]

Source: GMAT Prep

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