If-2f-x-f-1-x-x-2-determine-f-x-

Question Number 96829 by bobhans last updated on 05/Jun/20

If 2 f (x) + f (1 - x) = x^{2} . determine f (x)

Commented by john santu last updated on 05/Jun/20

replace x by 1 - x into eq (1)

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

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

multiply eq (1) by 2

(1) 2 f (1 - x) + 4 f (x) = {2 x}^{2}

subtract eq (2) by eq (1)

3 f (x) = {2 x}^{2} - (x^{2} - 2 x + 1)

3 f (x) = x^{2} + 2 x - 1

∴ f (x) = \frac{x^{2} + 2 x - 1}{3} .