For which of the following functions f is f(x) = f(1 - x) for all x?
(i) f(x)= 1-x
(ii) f(x)= 1-x²
(iii) f(x)= x²-(1-x)²
(iv) f(x)= x²(1-x)²
(v) f(x)= x/(1-x)

Solution:

Given function f(x) = f(1 – x)

By inspection,

(i) Consider f(x) = (1-x) then f(1-x) = 1-(1-x)

= x ≠ f(x)

(ii) Consider f(x) = 1-x² then f(1-x) = 1-(1-x)²

= 1-1+2x-x²

= 2x-x² ≠ f(x)

(iii) Consider f(x)= x²-(1-x)² then f(1-x)

= (1-x)² - [1-(1-x)]²

= 1-2x+x²-x²

= 1-2x ≠ f(x)

(iv) Consider f(x)= x²(1-x)² then f(1-x)

= (1-x)² [1 - (1-x)]² = (1-x)²x² = f(x)

(v) Consider f(x)= x/(1-x) then f(1-x)

= (1-x)/(1-(1-x))

= (1-x)/x ≠ f(x)

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For which of the following functions f is f(x) = f(1 - x) for all x?
(i) f(x)= 1-x
(ii) f(x)= 1-x²
(iii) f(x)= x²-(1-x)²
(iv) f(x)= x²(1-x)²
(v) f(x)= x/(1-x)

Summary:

For the given function f(x)= x²(1-x)², f(x) = f(1 – x) for all x.