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

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UpStudy AI · Accepted Answer

We start with the functions  
$$
f(x)=\sqrt{x-9} \quad \text{and} \quad g(x)=\sqrt{9-x}.
$$

**Step 1. Find the domains of $$ f $$ and $$ g $$.**

For $$ f(x)=\sqrt{x-9} $$, the expression under the square root must be non-negative:
$$
x-9 \ge 0 \quad \Rightarrow \quad x \ge 9.
$$
Thus, the domain of $$ f $$ is  
$$
\{x \mid x\ge 9\}.
$$

For $$ g(x)=\sqrt{9-x} $$, the expression under the square root must also be non-negative:
$$
9-x \ge 0 \quad \Rightarrow \quad x \le 9.
$$
Thus, the domain of $$ g $$ is  
$$
\{x \mid x\le 9\}.
$$

**Step 2. Find the domain of $$(f+g)(x)$$.**

The function $$(f+g)(x)= \sqrt{x-9}+\sqrt{9-x}$$ is defined only for values of $$ x $$ that belong to the domains of both $$ f $$ and $$ g $$. Therefore, the domain of $$(f+g)(x)$$ is the intersection:
$$
\{x\mid x\ge9\} \cap \{x\mid x\le9\} = \{9\}.
$$

**Step 3. Identify the correct choice.**

The correct answer is:  
A. The domain of $$ f+g $$ is $$ \{9\} $$.
The domain of $$ f+g $$ is $$ \{9\} $$.

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