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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!

Let f(x) = √(5x - 1) and g(x) = 1/x. Find (f + g)(x), (f - g)(x), (fg)(x), and (f/g)(x). Give the domain of each.

Learn how to calculate and find the domain of (f + g)(x), (f - g)(x), (fg)(x), and (f/g)(x) given f(x) = √(5x - 1) and g(x) = 1/x. This problem involves algebraic functions, square roots, and rational expressions, suitable for Grades 10-12.

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