First slide

Functions (XII)

Question

$The function f(x)= \frac{x + 1}{x^{3} + 1} c a n b e w r i t t e n a s t h e s u m o f a n e v e n f u n c t i o n g (x) a n d a n o d d f u n c t i o n h (x) . T h e n t h e v a l u e o f |g (0)| i s$

Moderate

Solution

$g(x)= \frac{f (x) + f (- x)}{2} = \frac{1}{2} [\frac{x + 1}{x^{3} + 1} + \frac{1 - x}{x^{3} + 1}] = \frac{1}{2} [\frac{1}{x^{2} - x + 1} + \frac{1}{1 + x + x^{2}}] = \frac{1}{2} [\frac{2 (x^{2} + 1)}{{(x^{2} + 1)}^{2} - x^{2}}] ∴ g (0) = = 1$

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