First slide

Algebra of real valued functions

Question

If $e^{f (x)} = \frac{10 + x}{10 - x}, x \in (- 10, 10)$ and $f (x) = k . f (\frac{200 x}{100 + x^{2}})$ then k = ______

Moderate

A

0.5
B

0.7
C

0.3
D

0.9

Solution

$e^{f (x)} = \frac{10 + x}{10 - x}, \Rightarrow f (x) = \log (\frac{10 + x}{10 - x}) f (x) = k . f (\frac{200 x}{100 + x^{2}}) = k \log (\frac{10 + \frac{200 x}{100 + x^{2}}}{10 - \frac{200 x}{100 + x^{2}}}) = k \log (\frac{10 (100 + x^{2}) + 200 x}{10 (100 + x^{2}) - 200 x}) = k \log (\frac{100 + x^{2} + 20 x}{100 + x^{2} - 20 x}) = k \log {(\frac{10 + x}{10 - x})}^{2}$
f(x) = 2k f(x)

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