The domain of definition of the function y (x) given by the equation `2^x +2^y = 2 `is

Options

A. `0 lt x lt 1`
B. ` 0 le x le 1`
C. ` - oo lt x le 0`
D. ` - oo lt x lt 1`

Answer

To find the domain of the function \( y(x) \) defined by the equation \( 2^x + 2^y = 2 \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ 2^x + 2^y = 2 \]
We can isolate \( 2^y \):
\[ 2^y = 2 - 2^x \]

### Step 2: Analyzing the Right Side
For \( 2^y \) to be defined, the right side \( 2 - 2^x \) must be greater than 0:
\[ 2 - 2^x > 0 \]
This simplifies to:
\[ 2^x < 2 \]

### Step 3: Solving the Inequality
The inequality \( 2^x < 2 \) can be rewritten using the properties of exponents:
\[ 2^x < 2^1 \]
Since the bases are the same, we can compare the exponents:
\[ x < 1 \]

### Step 4: Determining the Domain
The inequality \( x < 1 \) tells us that \( x \) can take any value less than 1. Thus, the domain of the function \( y(x) \) is:
\[ (-\infty, 1) \]

### Final Answer
The domain of the function \( y(x) \) is:
\[ x \in (-\infty, 1) \]