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GRE Prep Club Team Member
Joined: 20 Feb 2017
Posts: 2508
Given Kudos: 1053
GPA: 3.39
For which of the following functions f is f(x) = f(1-x) for all x?
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29 Nov 2022, 08:40
29 Nov 2022, 08:40
Expert Reply
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Question Stats:
75% (01:16) correct
25% (01:44) wrong based on 4 sessions
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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. \(f(x) = \frac{x}{(1 - x)}\)
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. \(f(x) = \frac{x}{(1 - x)}\)
GRE Prep Club Team Member
Joined: 20 Feb 2017
Posts: 2508
Given Kudos: 1053
GPA: 3.39
Re: For which of the following functions f is f(x) = f(1-x) for all x?
[#permalink]
30 Nov 2022, 22:42
30 Nov 2022, 22:42
1
Expert Reply
Okay, so you need to check if f(x) = f(1-x)
Let's do this one by one.
Case A: - Wrong
\(f(x) = 1-x\)
\(f(1-x) = 1-(1-x) = x\)
So we clearly see that f(x) is not f(1-x)
Case B: Wrong
\(f(x) = 1-x-x^2\)
\(f(1-x) = 1-(1-x)-(1-x)^2 = x - 1 - x^2 + 2x = 3x-1-x^2\)
Case C: Wrong
\(f(x) = x^2 - (1-x)^2 = x^2 - 1 +2x - x^2= 2x-1\)
\(f(1-x) = (1-x)^2 - (1-(1-x))^2 = (1-x)^2 - (x)^2= 1-2x\)
Case D: Right
\(f(x) = x^2 (1-x)^2\)
\(f(1-x) =(1-x)^2 (1-(1-x))^2= (1-x)^2(x)^2=f(x)\)
Let's check E just to be sure.
Case E: Wrong
\(f(x) = \frac{x}{1-x}\)
\(f(1-x) = \frac{1-x}{1-(1-x)}=\frac{1-x}{x}\)
Answer: D
Let's do this one by one.
Case A: - Wrong
\(f(x) = 1-x\)
\(f(1-x) = 1-(1-x) = x\)
So we clearly see that f(x) is not f(1-x)
Case B: Wrong
\(f(x) = 1-x-x^2\)
\(f(1-x) = 1-(1-x)-(1-x)^2 = x - 1 - x^2 + 2x = 3x-1-x^2\)
Case C: Wrong
\(f(x) = x^2 - (1-x)^2 = x^2 - 1 +2x - x^2= 2x-1\)
\(f(1-x) = (1-x)^2 - (1-(1-x))^2 = (1-x)^2 - (x)^2= 1-2x\)
Case D: Right
\(f(x) = x^2 (1-x)^2\)
\(f(1-x) =(1-x)^2 (1-(1-x))^2= (1-x)^2(x)^2=f(x)\)
Let's check E just to be sure.
Case E: Wrong
\(f(x) = \frac{x}{1-x}\)
\(f(1-x) = \frac{1-x}{1-(1-x)}=\frac{1-x}{x}\)
Answer: D
gmatclubot
Re: For which of the following functions f is f(x) = f(1-x) for all x? [#permalink]
30 Nov 2022, 22:42
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