We are given two functions $f(x) = \frac{1}{x-3}$ and $g(x) = \sqrt{2x+1}$. The task is to find:

1. $(f \times g)(x)$
2. The domain of $(f \times g)(x)$

a. $(f \times g)(x)$

The product of the two functions is the multiplication of $f(x)$ and $g(x)$:

$(f \times g)(x) = f(x) \cdot g(x) = \frac{1}{x-3} \cdot \sqrt{2x+1}$

Thus, the expression for $(f \times g)(x)$ is:

$(f \times g)(x) = \frac{\sqrt{2x+1}}{x-3}$

b. Domain of $(f \times g)(x)$

To find the domain, we must ensure that:

1. $f(x) = \frac{1}{x-3}$ is defined. This means $x-3 \neq 0$, so $x \neq 3$.
2. $g(x) = \sqrt{2x+1}$ is defined, which requires $2x + 1 \geq 0$, so $x \geq -\frac{1}{2}$.

The domain is the intersection of these two conditions:

$x \geq -\frac{1}{2} \quad \text{and} \quad x \neq 3$

Thus, the domain of $(f \times g)(x)$ is:

$\boxed{[-\frac{1}{2}, 3) \cup (3, \infty)}$

Would you like further clarification on any part? Here are some related questions:

1. How do you find the domain of a composite function?
2. What happens to the domain when a square root is involved?
3. How do you handle undefined values in rational functions?
4. What is the process to multiply two functions?
5. How can restrictions on domains affect function composition?

Tip: Always remember that when multiplying functions, consider the domains of both functions to ensure the result is valid for all values of $x$.