Question

if f(x) satisfies the relation 2f(x) +f(1-x)=x² for all real x then f(x) is​

Answer 1

$Given, 2f(x)+f(1−x)= {x}^{2} ...(i)$

Replacing x by (1−x), we get

$2f(1−x)+f(x)= {(1 - x)}^{2}$

$⇒ 2f(1−x)+f(x)=1+ {x}^{2} −2x ...(ii)$

On multiplying Eq. (i) by 2 and subtracting from Eq. (ii), we get

$3f(x)= {x}^{2} +2x−1$

$f(x)= {x}^{2} +2x−13$