First slide

Functions (XII)

Question

The domain of definition of the functions $y (x)$ given by the equation $a^{x} + a^{y} = a (a > 1)$ is

Moderate

A

$0 < x \leq 1$
B

$0 \leq x \leq 1$
C

$- \infty < x < 1$
D

$- \infty < x \leq 0$

Solution

$a^{x - 1} + a^{y - 1} = 1$
$\Rightarrow a^{y - 1} = 1 - a^{x - 1} \Rightarrow (y - 1) \log a = \log (1 - a^{x - 1})$
R.H.S. is defined if $1 - a^{x - 1} > 0$
$\Rightarrow a^{x - 1} < 1 = a^{\circ} \Rightarrow x - 1 < 0$ as $a > 1$
$\Rightarrow x < 1$ and the required domain is $- \infty < x < 1$ .

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