First slide

Types of real valued functions XI

Question

If $f (x) = \log (\frac{1 + x}{1 - x})$ ,then

Moderate

A

$f (x_{1}) \cdot f (x_{2}) = f (x_{1} + x_{2})$
B

$f (x + 2) - 2 f (x + 1) + f (x) = 0$
C

$f (x) + f (x + 1) = f (x^{2} + x)$
D

$f (x_{1}) + f (x_{2}) = f (\frac{x_{1} + x_{2}}{1 + x_{1} x_{2}})$

Solution

$\begin{array}{l} f (x_{1}) + f (x_{2}) = \log (\frac{1 + x_{1}}{1 - x_{1}} \cdot \frac{1 + x_{2}}{1 - x_{2}}) \\ = \log (\frac{1 + x_{1} x_{2} + x_{1} + x_{2}}{1 + x_{1} x_{2} - x_{1} - x_{2}}) \\ = \log (\frac{1 + \frac{x_{1} + x_{2}}{1 + x_{1} x_{2}}}{1 - \frac{x_{1} + x_{2}}{1 + x_{1} x_{2}}}) = f (\frac{x_{1} + x_{2}}{1 + x_{1} x_{2}}) \end{array}$

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