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)

i plugged in f[1-(1-x)] and got f(x). so i thought the answer was a. could someone show me how this works?

## function f(x)

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- neelgandham
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Easiest way to solve this PS is substitutionfangtray 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)

Let x = 1, then 1-x = 0, so f(1) = f(0)

a. f(x) = 1-x, f(1) = 0, f(0) = 1, f(x) is not equal to f(1-x), Incorrect

b. f(x) = 1-(x^2), f(1) = 0, f(0) = 1, f(x) is not equal to f(1-x), Incorrect

c. f(x) = (x^2)-(1-x)^2, f(1) = 1, f(0) = -1, f(x) is not equal to f(1-x), Incorrect

d. f(x) = (x^2)(1-x)^2, f(1) = 0, f(0) = 0, Correct!! Yippie! [spoiler]<-- answer[/spoiler]

e. f(x) = x/(1-x), f(1) = Not defined, f(0) = 0, Incorrect

Anil Gandham

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fangtray 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)

i plugged in f[1-(1-x)] and got f(x). so i thought the answer was a. could someone show me how this works?

Let us look at each of the options:

(A) f(x) = 1-x

f(1-x) = 1 - (1-x) = x; FALSE

(B) f(x) = 1-x^2

f(1-x) = 1 - (1-x)^2 = 1 - (1 - 2x + x^2) = 2x - x^2; FALSE

(C) f(x) = x^2 - (1-x)^2

f(1-x) = (1-x)^2 - (1 - (1-x))^2 = 1 - 2x + x^2 - (x)^2 = 1 - 2x; FALSE

(D) f(x) = x^2 * (1-x)^2

f(1-x) = (1 - x)^2 * (1 - (1 - x))^2 = (1 - x)^2 * (x)^2; TRUE

(E) f(x) = x/(1-x)

so f(1-x) = (1-x)/(1-(1-x)) = (1-x)/x; FALSE

The correct answer is D.

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a) f(x) = 1-x

==> f(1-x) = 1-(1-x)= x

hence f(x) not= f(1-x)

b)f(x) = 1-x^2

f(1-x) = 1-(1-x)^2

= 1-(1+(x^2)-2x)

= (-x^2)+2x

here, f(x) not=f(1-x)

c) f(x) = (x^2)-(1-x)^2

f(1-x)=(1-x)^2-(1-(1-x))^2

= (1-x)^2 -(x^2)

f(x) not = f(1-x), hence not the solution

d) f(x) = (x^2)(1-x)^2

f(1-x) = (1-x)^2.(1-(1-x))^2

= (1-x)^2.(x^2)

**Here, plz observe that f(x) and f(1-x) are the same. Hence option D is the answer.**