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 can be

Easy

A

2
B

$\frac{2}{3}$
C

$\frac{4}{3}$
D

$- \frac{1}{3}$

Solution

$\begin{array}{l} f (x) = \frac{a^{x} - 1}{x^{n} (a^{x} + 1)} \\ f (x) issymmetricalabout y - axis . Thus, \\ f (x) = f (- x) \end{array}$
or $\frac{a^{x} - 1}{x^{n} (a^{x} + 1)} = \frac{a^{x} - 1}{{(- x)}^{n} (a^{- x} + 1)}$
or $\frac{a^{x} - 1}{x^{n} (a^{x} + 1)} = \frac{1 - a^{x}}{{(- x)}^{n} (1 + a^{- x})}$
or $x^{n} = - {(- x)}^{n}$
Hence, the value of n which satisfies this relation is $- \frac{1}{3}$

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