Let's break down the given functions and solve the problems step by step.

You have:

• $f(x) = x + 5$
• $g(x) = x - 5$

We need to find:

(a) $(f + g)(x)$

This is the sum of the functions $f(x)$ and $g(x)$.

$(f + g)(x) = f(x) + g(x) = (x + 5) + (x - 5) = 2x$

(b) $(f - g)(x)$

This is the difference of the functions $f(x)$ and $g(x)$.

$(f - g)(x) = f(x) - g(x) = (x + 5) - (x - 5) = x + 5 - x + 5 = 10$

(c) $(fg)(x)$

This is the product of the functions $f(x)$ and $g(x)$.

$(fg)(x) = f(x) \cdot g(x) = (x + 5)(x - 5) = x^2 - 25$

(d) $\left(\frac{f}{g}\right)(x)$

This is the quotient of the functions $f(x)$ and $g(x)$.

$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x + 5}{x - 5}$

Domain of $\frac{f}{g}$

The domain of $\frac{f}{g}$ is all real numbers except where $g(x) = 0$, because division by zero is undefined.

$g(x) = x - 5 = 0 \quad \Rightarrow \quad x = 5$

Thus, the domain of $\frac{f}{g}$ is all real numbers except $x = 5$. In interval notation, this is:

$(-\infty, 5) \cup (5, \infty)$

Would you like more details or have any questions? Here are some related questions you could explore:

1. How do you find the domain of a rational function in general?
2. What happens if the domain of $f(x)$ and $g(x)$ intersect in problematic points (like division by zero)?
3. Can you explain how the difference of two functions is computed?
4. How would the domain change if $g(x) = x^2 - 25$ instead?
5. What is the graphical interpretation of adding or subtracting two functions?

Tip: When finding the domain of rational functions, always check where the denominator equals zero, as this will limit the domain.