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

Options

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

Answer

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

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

### Step 2: Analyzing the Right Side
For \(2^y\) to be valid (since \(2^y\) must be non-negative), the right side \(2 - 2^x\) must also be non-negative:
\[ 2 - 2^x \geq 0 \]
This implies:
\[ 2^x \leq 2 \]

### Step 3: Solving the Inequality
Now we solve the inequality \(2^x \leq 2\):
\[ 2^x \leq 2^1 \]
Taking the logarithm base 2 on both sides, we have:
\[ x \leq 1 \]

### Step 4: Defining the Domain
The inequality \(x \leq 1\) tells us that \(x\) can take any value less than or equal to 1. Therefore, the domain of the function \(y(x)\) is:
\[ (-\infty, 1] \]

### Conclusion
Thus, the domain of the function defined by the equation \(2^x + 2^y = 2\) is:
\[ \text{Domain: } (-\infty, 1] \]

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