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.
5-Day Free Trial
5-day free, full-access trial TTP
Available with Beat the GMAT members only code
MORE DETAILS
Help on PS problem- GMATprep
This topic has expert replies
-
gabriel
- Legendary Member
- Posts: 986
- Joined: Wed Dec 20, 2006 11:07 am
- Location: India
- Thanked: 51 times
- Followed by:1 members
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 wrote: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.
-
Cybermusings
- Legendary Member
- Posts: 559
- Joined: Tue Mar 27, 2007 1:29 am
- Thanked: 5 times
- Followed by:2 members
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
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
GMAT/MBA Expert
-
Scott@TargetTestPrep
- GMAT Instructor
- Posts: 7243
- Joined: Sat Apr 25, 2015 10:56 am
- Location: Los Angeles, CA
- Thanked: 43 times
- Followed by:29 members
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.Prasanna wrote: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) = 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
Scott Woodbury-Stewart
Founder and CEO
[email protected]
See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews
GMAT/MBA Expert
-
Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
Let's test the values using x = 0Prasanna wrote: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.
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
Brent Hanneson - Creator of GMATPrepNow.com
