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Let's go through each part of the problem step-by-step. We're given the functions:

\[
f(x) = \frac{7}{x} \quad \text{and} \quad g(x) = x - 6.
\]

We'll find each of the combined functions and their respective domains.

### 1. \( (f + g)(x) \)

\[
(f + g)(x) = f(x) + g(x) = \frac{7}{x} + (x - 6).
\]

#### Domain of \( (f + g)(x) \):

- For \( f(x) = \frac{7}{x} \) to be defined, \( x \neq 0 \) (since division by zero is undefined).
- \( g(x) = x - 6 \) is defined for all real numbers.

Therefore, the domain of \( (f + g)(x) \) is all real numbers except \( x = 0 \). In interval notation:

\[
D = (-\infty, 0) \cup (0, \infty).
\]

### 2. \( (f - g)(x) \)

\[
(f - g)(x) = f(x) - g(x) = \frac{7}{x} - (x - 6).
\]

#### Domain of \( (f - g)(x) \):

- The domain restrictions are the same as for \( (f + g)(x) \) since \( f(x) = \frac{7}{x} \) requires \( x \neq 0 \).

So, the domain of \( (f - g)(x) \) is also:

\[
D = (-\infty, 0) \cup (0, \infty).
\]

### 3. \( (fg)(x) \)

\[
(fg)(x) = f(x) \cdot g(x) = \frac{7}{x} \cdot (x - 6).
\]

#### Domain of \( (fg)(x) \):

- Again, \( f(x) = \frac{7}{x} \) requires \( x \neq 0 \).

Thus, the domain of \( (fg)(x) \) is:

\[
D = (-\infty, 0) \cup (0, \infty).
\]

### 4. \( \left(\frac{f}{g}\right)(x) \)

\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\frac{7}{x}}{x - 6}.
\]

#### Domain of \( \left(\frac{f}{g}\right)(x) \):

- \( f(x) = \frac{7}{x} \) requires \( x \neq 0 \).
- \( \frac{f}{g} \) requires that the denominator \( g(x) = x - 6 \neq 0 \). This means \( x \neq 6 \).

Combining these restrictions, the domain is all real numbers except \( x = 0 \) and \( x = 6 \). In interval notation:

\[
D = (-\infty, 0) \cup (0, 6) \cup (6, \infty).
\]

### Summary of the Solutions

1. **\( (f + g)(x) = \frac{7}{x} + (x - 6) \)**
   - Domain: \( (-\infty, 0) \cup (0, \infty) \).

2. **\( (f - g)(x) = \frac{7}{x} - (x - 6) \)**
   - Domain: \( (-\infty, 0) \cup (0, \infty) \).

3. **\( (fg)(x) = \frac{7}{x} \cdot (x - 6) \)**
   - Domain: \( (-\infty, 0) \cup (0, \infty) \).

4. **\( \left(\frac{f}{g}\right)(x) = \frac{\frac{7}{x}}{x - 6} \)**
   - Domain: \( (-\infty, 0) \cup (0, 6) \cup (6, \infty) \).

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What is the domain of the sum of two arbitrary functions \( f \) and \( g \)?
2. How do you determine the domain of a quotient of two functions?
3. What happens to the domain when you multiply two functions with different domains?
4. How would the domains change if \( f(x) \) or \( g(x) \) involved square roots or logarithms?
5. Can two functions have the same range but different domains?

**Tip:** Always consider the restrictions imposed by both functions when finding the domain of their combination.

Use the given functions f(x) = 7/x and g(x) = x - 6 to find each of the following functions and their respective domains. (f + g)(x), (f - g)(x), (fg)(x), (f / g)(x).

Learn how to find the sum, difference, product, and quotient of two functions f(x) = 7/x and g(x) = x - 6, along with their respective domains. This detailed breakdown covers key algebraic principles for high school students.

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