First slide

Functions (XII)

Question

$If the graph of the function f (x) = \frac{a^{x} - 1}{x^{n} (a^{x} + 1)} is symmetrical about the y -axis, then n equals$

Moderate

A

2
B

$\frac{2}{3}$
C

$\frac{1}{4}$
D

$- \frac{1}{3}$

Solution

$f (x) = \frac{a^{x} - 1}{x^{n} (a^{x} + 1)}$
$f (x) i s s y m m e t r i c a b o u t y - a x i s . T h u s, f (x) = f (- x)$
$\Rightarrow \frac{a^{x} - 1}{x^{n} (a^{x} + 1)} = \frac{a^{- x} - 1}{{(- x)}^{n} (a^{- x} + 1)}$
$= \frac{1 - a^{x}}{{(- x^{})}^{n} (1 + a^{x})}$
$\Rightarrow x^{n} = - {(- x)}^{n}$
$H e n c e, t h e v a l u e o f n w h i c h s a t i s f i e s t h i s r e l a t i o n i s - \frac{1}{3}$

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