Question

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

• A. $0\le x\le 2$
• B. $0\le x\le 1$
• C. $-\infty <x\le 0$
• D. $-\infty <x<1$

Answer 1

The correct option is D

$-\infty <x<1$

Explanation for the correct option:

Find the domain of $f\left(x\right)$:

Consider the given function as,

$a^x+a^y=a$

Rewrite the above Equation as, $a^y=a-a^x$

Take log on both side

$$\begin{gathered} \Rightarrow {\log }_a{a}^y={\log }_a\left(a-a^x\right) \\ \Rightarrow y={\log }_a\left(a-a^x\right) \end{gathered}$$

Then,

$$\begin{gathered} \Rightarrow a-a^x>0 \\ \Rightarrow a^x<0 \end{gathered}$$

Since, $a>0$ this happens only if the power $x$ is lesser than $1$

$$\begin{gathered} \Rightarrow x\in \left(-\infty ,1\right) \\ \Rightarrow -\infty <x<1 \end{gathered}$$

Hence, the domain of the function is $\left(-\infty ,1\right)$.

Therefore, the correct answer is option D.