First slide

Functions (XII)

Question

$Let f (x + \frac{1}{y}) + f (x - \frac{1}{y}) = 2 f (x) f (\frac{1}{y}) \forall x, y \in R$ $y \neq 0 and f (0) = 0 then the value of f (1) + f (2) =$

Moderate

A

-1
B

0
C

1
D

$none of these$

Solution

$Let f (x + \frac{1}{y}) + f (x - \frac{1}{y}) = 2 f (x) f (\frac{1}{y}) \forall x, y \in R$
$Given f (0) = 0$
$Putting x = 0, y = \frac{1}{x}, we get$
$\begin{aligned} f (x) + f (- x) = 2 f (0) f (x) \\ \Rightarrow f (x) + f (- x) = 0 \\ \Rightarrow f (x) = - f (- x) \end{aligned}$
$Putting x = 1, y = - 1, we get$
$\begin{array}{l} \Rightarrow f (2) + f (0) = 2 [f (1)]^{2} \\ \Rightarrow f (2) = 2 [f (1)]^{2} \end{array}$
$Putting x = - 1, y = - 1, we get$
$\begin{array}{l} \Rightarrow f (- 2) = 2 f (- 1) f (- 1) \\ \Rightarrow - f (2) = 2 (f (1))^{2} \\ \Rightarrow f (2) = - f (2) \\ \Rightarrow f (2) = 0 \\ \Rightarrow f (1) = 0 \\ ∴ f (1) = f (2) = 0 \\ ∴ f (1) + f (2) = 0 \end{array}$

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