First slide

Functions (XII)

Question

$If 3 f (x) + 5 f (\frac{1}{x}) = \frac{1}{x} - 3, \forall x (\neq 0) \in R$ , then f (x) =

Moderate

A

$\frac{1}{14} (\frac{3}{x} + 5 x - 6)$
B

$\frac{1}{14} (- \frac{3}{x} + 5 x - 6)$
C

$\frac{1}{14} (- \frac{3}{x} - 5 x - 6)$
D

None of these

Solution

We have,
$3 f (x) + 5 f (\frac{1}{x}) = \frac{1}{x} - 3, \forall x (\neq 0) \in R . . . . . . . (1) \Rightarrow 3 f (\frac{1}{x}) + 5 f (x) = x - 3 . . . . . . . (2)$
$[Replacing x by \frac{1}{x}]$
Multiplying Eq. (1) by 3 and (2) by 5 and subtracting, we get
$\begin{array}{l} 9 f (x) - 25 f (x) = (\frac{3}{x} - 9) - (5 x - 15) \\ \Rightarrow - 14 f (x) = \frac{3}{x} - 5 x + 6 \\ \Rightarrow f (x) = \frac{1}{14} (- \frac{3}{x} + 5 x - 6), \forall x (\neq 0) \in R \end{array}$

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