Question

The domain of definition of the function $y\left(x\right)$ is given by the equation $2^x+2^y=2$ is

• A. $0<x\le 1$
• 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:

We have the given equation as: $2^x+2^y=2$

Rearranging in terms of $y$, we get:

$\Rightarrow 2^y=2-2^x$

Taking $\log$ on both sides,

$$\begin{gathered} \Rightarrow \log \left(2^y\right)=\log (2-2^x) \\ \Rightarrow y\log \left(2\right)=\log (2-2^x)[\because \log (a^x)=x\log (a\left)\right] \end{gathered}$$

We know, that the log value is defined only for the positive values,

$$\begin{gathered} \therefore 2-2^x>0 \\ \Rightarrow 2>2^x \\ \Rightarrow 2^x<2^1 \end{gathered}$$

Comparing the exponents we have, $x<1$, as for $x\ge 1$, the inequality will hold false.

Therefore, we have the domain of $x$ as: $(-\infty ,1)$ or $-\infty <x<1$

Hence, the correct answer is option (D)