$$ (\mathrm{f}-\mathrm{g})(\mathrm{x})=\sqrt{\mathrm{x}-9}-\sqrt{9-\mathrm{x}} \mathrm{s} $$ What is the domain of $$ \mathrm{f}-\mathrm{g} $$ ? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain of $$ \mathrm{f}-\mathrm{g} $$ is $$ \{ $$. (Use a comma to separate answers as needed.) B. The domain of $$ \mathrm{f}-\mathrm{g} $$ is (Type your answer in interval notation.) C . The domain of $$ \mathrm{f}-\mathrm{g} $$ is $$ \varnothing $$.

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UpStudy AI · Accepted Answer

To find the domain of the function $$ (\mathrm{f}-\mathrm{g})(\mathrm{x})=\sqrt{\mathrm{x}-9}-\sqrt{9-\mathrm{x}} $$, we need to determine the values of $$ x $$ for which this expression is defined.

1. **Identify the conditions for the square roots:**
   - The expression $$ \sqrt{\mathrm{x}-9} $$ is defined when $$ \mathrm{x}-9 \geq 0 $$, which simplifies to:
     $$
     \mathrm{x} \geq 9
     $$
   - The expression $$ \sqrt{9-\mathrm{x}} $$ is defined when $$ 9-\mathrm{x} \geq 0 $$, which simplifies to:
     $$
     \mathrm{x} \leq 9
     $$

2. **Combine the conditions:**
   - From the first condition, we have $$ \mathrm{x} \geq 9 $$.
   - From the second condition, we have $$ \mathrm{x} \leq 9 $$.
   - The only value that satisfies both conditions is $$ \mathrm{x} = 9 $$.

3. **Conclusion:**
   - Therefore, the domain of $$ \mathrm{f}-\mathrm{g} $$ is the single point $$ \{ 9 \} $$.

Now, let's express this in the required format:

**Choice A:** The domain of $$ \mathrm{f}-\mathrm{g} $$ is $$ \{ 9 \} $$.
The domain of $$ \mathrm{f}-\mathrm{g} $$ is $$ \{9\} $$.

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