To find the domain of the function $(f \cdot g)(x)$, we need to consider the domain restrictions of both functions individually, then combine them. Here's the step-by-step process:

1. Domain of $f(x) = \sqrt{x - 5}$

• The square root function is only defined when the expression inside the root is non-negative.
• So, $x - 5 \geq 0$.
• Solving this inequality: $x \geq 5$
• Therefore, the domain of $f(x)$ is $[5, \infty)$.

2. Domain of $g(x) = \sqrt{-x + 5}$

• Again, the square root function is only defined when the expression inside the root is non-negative.
• So, $-x + 5 \geq 0$, or equivalently $x \leq 5$.
• Therefore, the domain of $g(x)$ is $(-\infty, 5]$.

3. Domain of $(f \cdot g)(x)$

• The domain of $(f \cdot g)(x)$ is the intersection of the domains of $f(x)$ and $g(x)$, because both functions need to be defined at the same time for their product to exist.
• The domain of $f(x)$ is $[5, \infty)$ and the domain of $g(x)$ is $(-\infty, 5]$.
• The intersection of these two domains is the single point $x = 5$.

Final Answer:

The domain of $(f \cdot g)(x)$ is $\{5\}$.

Would you like further details or have any other questions?

Here are 5 related questions to deepen your understanding:

1. What is the range of $(f \cdot g)(x)$ given the domain?
2. What happens to the domain if the functions are added instead of multiplied?
3. Can $f(x)$ and $g(x)$ ever be negative for any real number?
4. What is the value of $(f \cdot g)(x)$ at $x = 5$?
5. How would the domain change if $f(x) = \sqrt{x + 5}$?

Tip: Always check the domain restrictions imposed by the function types (e.g., square roots require non-negative values inside).