Find domain of the function `10^x+10^y=10`

Answer

To find the domain of the function given by the equation \(10^x + 10^y = 10\), we will analyze the conditions under which this equation holds true.

### Step-by-Step Solution:

1. **Rearranging the Equation**:
We start with the equation:
\[ 10^x + 10^y = 10 \]
We can rearrange it as:
\[ 10^y = 10 - 10^x \]

2. **Finding Conditions for \(10^y\)**:
For \(10^y\) to be defined and positive, the right-hand side must be greater than 0:
\[ 10 - 10^x > 0 \]
This simplifies to:
\[ 10^x < 10 \]

3. **Solving the Inequality**:
We can rewrite the inequality:
\[ 10^x < 10^1 \]
Taking logarithm base 10 on both sides gives:
\[ x < 1 \]

4. **Finding the Range of \(y\)**:
Now, substituting \(x < 1\) back into our rearranged equation, we can find \(y\):
\[ 10^y = 10 - 10^x \]
Since \(10^x\) is positive and less than 10, \(10 - 10^x\) will also be positive. Thus, \(y\) can take any real value as long as \(10^y\) remains defined.

5. **Conclusion on the Domain**:
Therefore, the domain of the function is:
\[ x < 1 \quad \text{and} \quad y \in \mathbb{R} \]
In interval notation, the domain can be expressed as:
\[ (-\infty, 1) \times \mathbb{R} \]

### Final Answer:
The domain of the function \(10^x + 10^y = 10\) is:
\[ \{(x, y) \mid x < 1, y \in \mathbb{R}\} \]

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