Question

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

• A. $0<x\le 1$
• B. $0$ $\le$ $x<1$
• C. $-\infty <x<1$
• D. $-\infty <x\le 0$.

Answer 1

The correct option is C $-\infty <x<1$

$a^x+a^y=a\Rightarrow a^y=a-a^x\Rightarrow y={\log }_a(a-a^x)$

$\Rightarrow a-a^x>0$ (Since logarithm of a negative number do not exist)

$\Rightarrow a^x<a$

This happen only if the power of x is less than $1$ (Since a>1)

$\Rightarrow x\in (-\infty ,1)$

$\Rightarrow -\infty <x<1\left[C\right]$