First slide

Algebra of real valued functions

Question

If for nonzero x, $a f (x) + b f (\frac{1}{x}) = \frac{1}{x} - 5.$ where $a \neq b,$ then f(2) =

Moderate

A

$\frac{3 (2 b + 3 a)}{2 (a^{2} - b^{2})}$
B

$\frac{3 (2 b - 3 a)}{2 (a^{2} - b^{2})}$
C

$\frac{3 (3 a - 2 b)}{2 (a^{2} - b^{2})}$
D

$\frac{6}{a + b}$

Solution

$a f (x) + b f (\frac{1}{x}) = \frac{1}{x} - 5 . . . . . . . . . . . (1) p u t x = 1 / x \Rightarrow a f (\frac{1}{x}) + b f (x) = x - 5 . . . . . . . . . . (2) a (1) - b (2) \Rightarrow (a^{2} - b^{2}) f (x) = \frac{a}{x} - b x - 5 (a - b)$
put x=2

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