Help on PS problem- GMATprep

[Prasanna]

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)=x/(1-x)

The answer is d. Need help in solving this.

[gabriel]

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)=x/(1-x)

The answer is d. Need help in solving this.

Hi prassana .... when we say f(x)=f(1-x) ... it means that the eqn that represents f(x) should not change when x is replaced with (1-x) in that eqn .. and if u see it is only true for d ... for d it is given f(x) = x^2(1-x)^2 .. to find f(1-x) replace x with 1-x ... so we have f(1-x)= (1-x)^2(1-(1-x))^2 = (1-x)^2x^2 ... which is the same as f(x).. so D ....

[Prasanna]

Gabriel

Thanks for the answer.

I had made a mistake in my calculations

Prasanna

[Cybermusings]

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)=x/(1-x)

a) f(x) = 1-x
f(1-x) = 1-1+x = x
Hence f(x) is not equal to f(1-x)

b) f(x) = [1-x^2]
f(1-x) = [1-(1-x)^2]
= [1-(1+x^2-2)]
=[2-x^2]
Hence f(x) is not equal to f(1-x)

c) f(x) = x^2-(1-x)^2
=x^2 - [ 1+x^2-2x]
=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 + x^2 - 2x - x^2
= 1-2x

d)f(x)=x^2(1-x)^2
f(x) = x^2(1+x^2-2x)
=x^2+x^4-2x^3

f(1-x) =x^2(1-x)^2
=[(1-x)^2] [ 1-(1-x)]^2
=[1+x^2-2x] [x^2]
=x^2 + x^4 - 2x^3

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

Hence D

[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)=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

[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)=x/(1-x)

The answer is d. Need help in solving this.

Let's 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