Question

Find f(x) if it satisfies the relation

$2f\left(x\right)+f(1-x)=x^2$

• A. $\frac{x^2+2x-1}{3}$
• B. $\frac{x^2+4x-1}{6}$
• C. $\frac{x^2+4x-1}{3}$
• D. $\frac{x^2+2x+1}{6}$

Answer 1

The correct option is A

$\frac{x^2+2x-1}{3}$

$2f\left(x\right)+f(1-x)=x^2....\left(1\right)$

Replacing x with (1-x), we get

$2f(1-x)+f\left(x\right)=(1-x{)}^2....(2)$

$2\times \left(1\right)-\left(2\right)\Rightarrow 3f\left(x\right)=2x^2-(1-x{)}^2$

$\Rightarrow f\left(x\right)=\frac{x^2+2x-1}{3}$