Question

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

• A. $\lim_{x \rightarrow \infty}\frac{1 - (2\sin x)^{2n}}{1 + (2\sin x)^{2n}}$
• B. x² – 2
• C. 1
• D. None of these

Answer 1

Answer: (A)

$\lim_{x \rightarrow \infty}\frac{1 - (2\sin x)^{2n}}{1 + (2\sin x)^{2n}}$

Answer 2

By replacing x with (1 – x) in the given expression, we get

f(1 – x) + 2 f(1- 1 + x) = (1 – x)² + 2 ⇒ f(1 – x) + 2 f(x)

= (1 - x)² + 2

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

((1 – x)²+2)

⇒ -3 f(x) = x² + 2 – 2(3 – 2x + x²)

⇒ 3 f(x) = x² – 4x + 4 ⇒ f(x) = $\frac { ( x - 2 ) ^ { 2 } } { 3 }$