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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-x^{2}

(iii) f(x)= x^{2}-(1-x)^{2}

(iv) f(x)= x^{2}(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-x^{2} then f(1-x) = 1-(1-x)^{2}

= 1-1+2x-x^{2}

= 2x-x^{2} ≠ f(x)

(iii) Consider f(x)= x^{2}-(1-x)^{2} then f(1-x)

= (1-x)^{2} - [1-(1-x)]^{2}

= 1-2x+x^{2}-x^{2}

= 1-2x ≠ f(x)

(iv) Consider f(x)= x^{2}(1-x)^{2} then f(1-x)

= (1-x)^{2} [1 - (1-x)]^{2} = (1-x)^{2}x^{2} = f(x)

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

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

= (1-x)/x ≠ f(x)

## 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-x^{2}

(iii) f(x)= x^{2}-(1-x)^{2}

(iv) f(x)= x^{2}(1-x)^{2}

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

**Summary:**

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

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