If f x -2 f 1- x - x 2-1 - x - R then f x isA- 1-3x2-4 x-3B- 2-3x2-4 x-3- 1-3x2-4 x-3D- 2-3x2-4 x-3

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Domain

If f x +2 f 1...

Question

If $f (x) + 2 f (1 - x) = x^{2} + 1 \forall x ϵ R$ then f(x) is

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Solution

The correct option is C
$f (x) + 2 f (1 - x) = x^{2} + 1$ .........(i)
Replacing x by 1 – x
$f (1 - x) + 2 f (x) = (1 - x^{2}) + 1$ .......(ii)
multiplying (ii) by 2 and subtracting it from (1), we get
$- 3 f (x) = x^{2} - 2 (1 - x)^{2} - 1$
$3 f (x) = 2 (1 - x)^{2} + 1 - x^{2} = (1 - x) (2 - 2 x + 1 + x) = (1 - x) (3 - x) = x^{2} - 4 x + 3$
$f (x) = \frac{1}{3} (x^{2} - 4 x + 3)$

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If f(x)+2f(1−x)=x2+1 ∀ xϵR then f(x) is

The correct option is C f(x)+2f(1−x)=x2+1 .........(i) Replacing x by 1 – x f(1−x)+2f(x)=(1−x2)+1 .......(ii) multiplying (ii) by 2 and subtracting it from (1), we get −3f(x)=x2−2(1−x)2−1 3f(x)=2(1−x)2+1−x2=(1−x)(2−2x+1+x)=(1−x)(3−x)=x2−4x+3 f(x)=13(x2−4x+3)

If $f (x) + 2 f (1 - x) = x^{2} + 1 \forall x ϵ R$ then f(x) is