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We are given two functions, \( f(x) = x - 3 \) and \( g(x) = \sqrt{3x - 4} \), and we need to find:

1. The function \( \frac{f}{g} \)
2. The function \( f - g \)

Additionally, we need to determine the domain of each resulting function using interval notation.

### 1. Finding \( \frac{f}{g} \):
\[
\frac{f}{g}(x) = \frac{x - 3}{\sqrt{3x - 4}}
\]

#### Domain of \( \frac{f}{g} \):
For the domain of \( \frac{f}{g} \), we need to consider two things:
- The square root function \( \sqrt{3x - 4} \) requires that \( 3x - 4 \geq 0 \), so:
  \[
  3x \geq 4 \implies x \geq \frac{4}{3}
  \]
- Additionally, since \( g(x) = \sqrt{3x - 4} \) is in the denominator, we must ensure that \( g(x) \neq 0 \). Therefore:
  \[
  \sqrt{3x - 4} \neq 0 \implies 3x - 4 \neq 0 \implies x \neq \frac{4}{3}
  \]
Thus, the domain of \( \frac{f}{g} \) is \( \left( \frac{4}{3}, \infty \right) \).

### 2. Finding \( f - g \):
\[
f - g(x) = (x - 3) - \sqrt{3x - 4}
\]

#### Domain of \( f - g \):
For the domain of \( f - g \), we only need to ensure that the square root \( \sqrt{3x - 4} \) is defined, which means:
\[
3x - 4 \geq 0 \implies x \geq \frac{4}{3}
\]
So, the domain of \( f - g \) is \( \left[ \frac{4}{3}, \infty \right) \).

### Final Results:

1. The function \( \frac{f}{g}(x) = \frac{x - 3}{\sqrt{3x - 4}} \) has a domain of \( \left( \frac{4}{3}, \infty \right) \).
2. The function \( f - g(x) = (x - 3) - \sqrt{3x - 4} \) has a domain of \( \left[ \frac{4}{3}, \infty \right) \).

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Would you like more details on the steps? Here are some related questions:

1. What happens if \( g(x) \) equals zero in more general cases?
2. How do you simplify rational expressions like \( \frac{f}{g} \)?
3. Can \( f - g(x) \) ever be negative? Why or why not?
4. What are the general rules for finding domains when square roots are involved?
5. How does the difference in domain intervals between \( \frac{f}{g} \) and \( f - g \) affect their graphs?

**Tip**: Always check for restrictions like square roots and denominators when determining domains.

Suppose that the functions f and g are defined as follows: f(x) = x - 3 and g(x) = √(3x - 4). Find f/g and f - g. Then, give their domains using interval notation.

Learn how to solve for f/g and f - g given f(x) = x - 3 and g(x) = √(3x - 4). The solution includes determining the domains of both functions using interval notation and covers key algebraic concepts such as square roots and rational expressions. Suitable for Grades 10-12.

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