First slide

Functions (XII)

Question

$If a f (x) + b f (\frac{1}{x}) = x - 1, x \neq 0, a \neq b, then f (2) =$

Moderate

A

$\frac{a + 2 b}{2 (a^{2} - b^{2})}$
B

$\frac{2 a + b}{a^{2} - b^{2}}$
C

$\frac{a + 2 b}{a^{2} - b^{2}}$
D

$\frac{2 a + b}{2 (a^{2} - b^{2})}$

Solution

$\begin{array}{l} Given that a f (x) + b f (\frac{1}{x}) = x - 1 \\ Replacing x by \frac{1}{x} \\ \Rightarrow b f (x) + a f (\frac{1}{x}) = \frac{1}{x} - 1 \\ Eliminating f (\frac{1}{x}) \\ we get (a^{2} - b^{2}) f (x) = a (x - 1) - b (\frac{1}{x} - 1) \\ \Rightarrow (a^{2} - b^{2}) f (2) = a + \frac{b}{2} \\ ∴ f (2) = \frac{2 a + b}{2 (a^{2} - b^{2})} \end{array}$

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