Question

plz solve it. plzzzzz

Answer 1

$\sf{Given : f(x) + 2f(1 - x) = x^2 + 2}$

$\textsf{Take the above Equation as Equation [1]}$

$\sf{\implies f(x) + 2f(1 - x) = x^2 + 2\;------\;[1]}$

$\textsf{Substituting (1 - x) in place of x in Equation [1], We get :}$

$\sf{\implies f(1 - x) + 2f\big(1 - (1 - x)\big) = (1 - x)^2 + 2}$

$\sf{\implies f(1 - x) + 2f\big(1 - 1 + x\big) = 1 + x^2 - 2x + 2}$

$\sf{\implies f(1 - x) + 2f(x) = x^2 - 2x + 3}$

$\textsf{Multiplying the Above Equation with 2, We get :}$

$\sf{\implies 2f(1 - x) + 4f(x) = 2x^2 - 4x + 6}\;------\;[2]$

$\textsf{Subtracting Equation [1] from Equation [2], We get :}$

$\sf{\implies 2f(1 - x) + 4f(x) - [f(x) + 2f(1 - x)] = 2x^2 - 4x + 6 - [x^2 + 2]}$

$\sf{\implies 2f(1 - x) + 4f(x) - f(x) - 2f(1 - x) = 2x^2 - 4x + 6 - x^2 - 2}$

$\sf{\implies 3f(x) = x^2 - 4x + 4}$

$\sf{\implies f(x) = \bigg(\dfrac{x^2 - 4x + 4}{3}\bigg)}$