COMEDK · Maths · 24. Functions
If \(2 f\left(x^2\right)+3 f\left(\dfrac{1}{x^2}\right)=x^2-1, \forall x \in R-\{0\}\), then \(f\left(x^8\right)\) is equal to
- A \(\dfrac{\left(1+x^8\right)\left(2 x^8-3\right)}{5 x^8}\)
- B \(\dfrac{\left(1-x^8\right)\left(2 x^8-3\right)}{5 x^8}\)
- C \(\dfrac{\left(1-x^8\right)\left(2 x^8+3\right)}{5 x^8}\)
- D None of these
Answer & Solution
Correct Answer
(C) \(\dfrac{\left(1-x^8\right)\left(2 x^8+3\right)}{5 x^8}\)
Step-by-step Solution
Detailed explanation
Given the equation \(2 f(x^2) + 3 f\left(\dfrac{1}{x^2}\right) = x^2 - 1\). Let \(t = x^2\). Then the equation becomes \(2 f(t) + 3 f\left(\dfrac{1}{t}\right) = t - 1\) for all \(t > 0\). Replacing \(t\) with \(\dfrac{1}{t}\), we get…
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