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The exercise involves combining two functions \( f(x) \) and \( g(x) \) in various ways:

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

Each numbered pair defines a function \( f(x) \) and a function \( g(x) \), and you need to determine the result of combining them using the above operations.

Let's go through the first example in detail to clarify the process:

### 1. Given:
- \( f(x) = \sqrt{x + 1} \)
- \( g(x) = \sqrt{x - 4} \)

For this pair:
- \( (f + g)(x) = \sqrt{x+1} + \sqrt{x-4} \)
- \( (f - g)(x) = \sqrt{x+1} - \sqrt{x-4} \)
- \( (f \cdot g)(x) = \sqrt{x+1} \cdot \sqrt{x-4} = \sqrt{(x+1)(x-4)} \)
- \( \left( \frac{f}{g} \right)(x) = \frac{\sqrt{x+1}}{\sqrt{x-4}} = \sqrt{\frac{x+1}{x-4}} \)

Would you like me to continue with the other exercises or explain any of these steps in more detail? 

Here are 5 related questions to deepen your understanding:
1. What are the conditions on the domain for each function involving square roots?
2. How would the product of two functions with square roots affect the resulting function's domain?
3. In which cases would division between two functions become undefined?
4. How do the results change if you replace a square root with another function form?
5. Can we always simplify the expression \( \sqrt{(x+1)(x-4)} \)?

**Tip**: Always check the domain restrictions before applying operations to functions, especially with square roots or fractions.

Ejercicios propuestos de álgebra de funciones. Determine las funciones resultantes para las combinaciones (f + g)(x), (f - g)(x), (f · g)(x), and (f / g)(x) using given f(x) and g(x) pairs.

Learn how to combine functions f(x) and g(x) using addition, subtraction, multiplication, and division in algebra. Explore key concepts, formulas, and domain restrictions with examples of square root and rational expressions.

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