Let’s go through each of the functions and compute their operations step by step, along with determining their domain.

Part 1: (f + g)(x)

Given:

• $f(x) = 11 - \frac{132}{x + 11}$
• $g(x) = \frac{11}{x}$

To compute $(f + g)(x) = f(x) + g(x)$, we first write it as a single fraction:

$(f + g)(x) = \left( 11 - \frac{132}{x + 11} \right) + \frac{11}{x}$

Now, let's combine these two terms into a single fraction. To do that, we need a common denominator. The denominators are $(x + 11)$ and $x$, so the common denominator is $x(x + 11)$. Let’s rewrite each term with this common denominator:

$f(x) = 11 - \frac{132}{x + 11} = \frac{11x(x + 11)}{x(x + 11)} - \frac{132}{x + 11} = \frac{11x^2 + 121x}{x(x + 11)} - \frac{132}{x + 11}$ $g(x) = \frac{11}{x} = \frac{11(x + 11)}{x(x + 11)}$

Now, add the two fractions:

$(f + g)(x) = \frac{11x^2 + 121x - 132 + 11(x + 11)}{x(x + 11)}$

Simplify the numerator:

$11x^2 + 121x - 132 + 11x + 121 = 11x^2 + 132x - 11$

So, we get:

$(f + g)(x) = \frac{11x^2 + 132x - 11}{x(x + 11)}$

Thus, the simplified expression for $(f + g)(x)$ is:

$(f + g)(x) = \frac{11x^2 + 132x - 11}{x(x + 11)}$

Part 2: Domain of $(f + g)(x)$

To determine the domain of $(f + g)(x)$, we must find the values of $x$ where the expression is undefined. The expression is undefined when either of the denominators $x$ or $x + 11$ equals zero.

• $x = 0$ makes the denominator zero.
• $x = -11$ makes the denominator zero.

Therefore, the domain is all real numbers except $x = 0$ and $x = -11$, which in interval notation is:

$(-\infty, -11) \cup (-11, 0) \cup (0, \infty)$

Summary:

• (f + g)(x) is $\frac{11x^2 + 132x - 11}{x(x + 11)}$
• The domain of $(f + g)(x)$ is $(-\infty, -11) \cup (-11, 0) \cup (0, \infty)$.

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Would you like further details on any part of this solution or help with parts b, c, d, and e? Here are a few questions that might interest you:

1. How do we find the domain of a function with multiple terms?
2. What happens if we try to evaluate $f(x)$ and $g(x)$ at $x = -11$ or $x = 0$?
3. How would you simplify similar expressions for other functions?
4. How do operations like addition, subtraction, multiplication, and division affect the domain?
5. What are some common techniques for combining rational functions?

Tip: When combining rational functions, always check for common denominators and simplify the result wherever possible.