Sure, let's plug the functions $f(x) = x^2$ and $g(x) = x - 5$ into the four operations and simplify the results:

1. Sum of the functions: $(f + g)(x)$

The sum of two functions is given by: $(f + g)(x) = f(x) + g(x)$ So, plugging in the given functions: $(f + g)(x) = x^2 + (x - 5)$ Simplify: $(f + g)(x) = x^2 + x - 5$

2. Product of the functions: $(f \cdot g)(x)$

The product of two functions is given by: $(f \cdot g)(x) = f(x) \cdot g(x)$ So, plugging in the given functions: $(f \cdot g)(x) = x^2 \cdot (x - 5)$ Distribute: $(f \cdot g)(x) = x^3 - 5x^2$

3. Difference of the functions: $(f - g)(x)$

The difference of two functions is given by: $(f - g)(x) = f(x) - g(x)$ So, plugging in the given functions: $(f - g)(x) = x^2 - (x - 5)$ Simplify: $(f - g)(x) = x^2 - x + 5$

4. Sum of the functions (again): $(f + g)(x)$

This is the same operation as the first one: $(f + g)(x) = x^2 + x - 5$ (Same result as before.)

Summary of Results:

1. $(f + g)(x) = x^2 + x - 5$
2. $(f \cdot g)(x) = x^3 - 5x^2$
3. $(f - g)(x) = x^2 - x + 5$
4. $(f + g)(x) = x^2 + x - 5$ (Same as the first)

If you'd like any more details or further simplification, feel free to ask!

---

Additional questions for you:

1. What happens if we divide the functions $f(x)$ and $g(x)$? (i.e., $(f / g)(x)$)
2. Can we apply these same operations for other types of functions (e.g., trigonometric or exponential)?
3. How would these operations affect the behavior of a graph, especially in terms of its shape?
4. How does the operation $f(g(x))$ differ from $g(f(x))$?
5. Would any of these operations be undefined for certain values of $x$? (e.g., division by zero)

Tip: When adding, subtracting, or multiplying functions, always pay attention to the order and ensure proper distribution in the case of products.