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Question

Question: If f(x) + 2f(1 - x) = x<sup>2</sup> +1 ∀ x ∈ R then f(x) is...

If f(x) + 2f(1 - x) = x2 +1 ∀ x ∈ R then f(x) is

A

13\frac{1}{3} (x2 + 4x - 3)

B

23\frac{2}{3} (x2 + 4x – 3)

C

13\frac{1}{3} (x2 – 4x + 3)

D

23\frac{2}{3} (x2 – 4x + 3)

Answer

13\frac{1}{3} (x2 – 4x + 3)

Explanation

Solution

f(x) + 2f(1 - x) = x2 + 1

Replacing x → (1 - x) we get; f(1 – x) + 2f(x) = (1 -x)2+ 1

⇒ f(1 - x) + 2f(x) = 2 + x2 - 2x

From these equations we get f(x) = 13\frac { 1 } { 3 } (x2 - 4x + 3)