We are given two functions:

• $f(x) = \sqrt{5x - 1}$
• $g(x) = \frac{1}{x}$

We are tasked with finding the following:

1. $(f + g)(x) = f(x) + g(x)$
2. $(f - g)(x) = f(x) - g(x)$
3. $(fg)(x) = f(x) \cdot g(x)$
4. $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$

Additionally, we need to determine the domain for each case.

1. $(f + g)(x) = f(x) + g(x)$

Using the given expressions for $f(x)$ and $g(x)$: $(f + g)(x) = \sqrt{5x - 1} + \frac{1}{x}$

Domain:

• For $f(x) = \sqrt{5x - 1}$, the expression inside the square root must be non-negative, i.e. $5x - 1 \geq 0 \Rightarrow x \geq \frac{1}{5}$.
• For $g(x) = \frac{1}{x}$, $x \neq 0$.

Thus, the domain of $(f + g)(x)$ is $x \in \left(\frac{1}{5}, \infty \right)$, excluding $0$ because $g(x)$ is undefined at $x = 0$.

Let me know if you'd like to proceed with the next parts or if you have any questions about this one!

Here are some relative questions you could try:

1. What is the result of $(f - g)(x)$?
2. How do you find the product of two functions, like $f(x)$ and $g(x)$?
3. What happens when you divide functions like $\frac{f(x)}{g(x)}$?
4. Can you explain how to determine the domain of a function involving a square root and a rational term?
5. How does the domain change when combining functions like $f(x) + g(x)$?

Tip: Always look at the conditions for each function separately when finding the domain of a combination of functions!