First slide

Functions (XII)

Question

$Let G (x) = (\frac{1}{a^{x} - 1} + \frac{1}{2}) F (x), where a is a positive real$ $number not equal to 1 and F (x) is an odd function. Which$ $of the following statements is true?$

Moderate

A

$G (x) is an odd function$
B

$G (x) is an even function$
C

$G (x) is neither even function nor odd function$
D

$Whether G (x) is an odd function or an even function, it depends on the value of a$

Solution

$\begin{array}{l} G (x) = (\frac{1}{a^{x} - 1} + \frac{1}{2}) F (x) \\ ∴ G (- x) = (\frac{1}{a^{- x} - 1} + \frac{1}{2}) F (- x) \\ = - (\frac{a^{x}}{1 - a^{x}} + \frac{1}{2}) F (x) \\ = (\frac{a^{x}}{a^{x} - 1} - \frac{1}{2}) F (x) \\ = (\frac{a^{x} - 1 + 1}{a^{x} - 1} - \frac{1}{2}) F (x) \\ = (1 + \frac{1}{a^{x} - 1} - \frac{1}{2}) F (x) \\ = (\frac{1}{a^{x} - 1} + \frac{1}{2}) F (x) = G (x) \end{array}$
$∴ G (x) is an even function.$

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