First slide

Functions (XII)

Question

The function $f (x) = \frac{x + 1}{x^{3} + 1}$ can be written as the sum of an even function g(x) and an odd function h(x). Then the value of $|g (0)|$ is _________

Moderate

Solution

$\begin{array}{l} g (x) = \frac{f (x) + f (- x)}{2} \\ = \frac{1}{2} [\frac{1}{x^{2} - x + 1} + \frac{1}{1 + x + x^{2}}] \\ = \frac{1}{2} [\frac{2 (x^{2} + 1)}{{(x^{2} + 1)}^{2} - x^{2}}] \\ = \frac{x^{2} + 1}{x^{4} + x^{2} + 1} \\ ∴ g (0) = 1 \end{array}$

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