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The image shows a question where you are given two functions:

- \( f(x) = 5x - 9 \)
- \( g(x) = x - 7 \)

The tasks are to:

1. Find \( \left( \frac{f}{g} \right)(x) \), and determine its domain.
2. Find \( \left( f \circ g \right)(x) \), and determine its domain.

Let's solve these step by step:

### (a) \( \left( \frac{f}{g} \right)(x) \)
This is the division of the two functions, \( \frac{f(x)}{g(x)} \).

\[
\frac{f(x)}{g(x)} = \frac{5x - 9}{x - 7}
\]

The domain for a rational function is all real numbers except where the denominator is zero. So, we need to find where \( g(x) = 0 \):

\[
x - 7 = 0 \implies x = 7
\]

Thus, the domain is all real numbers except \( x = 7 \). In interval notation, the domain is:

\[
(-\infty, 7) \cup (7, \infty)
\]

### (b) \( (f \circ g)(x) \)
This is the composition of functions, \( f(g(x)) \). To find this, substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f(x - 7) = 5(x - 7) - 9 = 5x - 35 - 9 = 5x - 44
\]

The domain of \( f \circ g \) is the domain of \( g(x) \), which is all real numbers since there are no restrictions on \( x \) in \( g(x) = x - 7 \).

So, the domain of \( f \circ g \) is all real numbers, or \( (-\infty, \infty) \).

### Final Answers:

- \( \left( \frac{f}{g} \right)(x) = \frac{5x - 9}{x - 7} \)
  - Domain: \( (-\infty, 7) \cup (7, \infty) \)
  
- \( (f \circ g)(x) = 5x - 44 \)
  - Domain: \( (-\infty, \infty) \)

Do you have any questions or would you like further details?

Here are five related questions to expand on this topic:

1. How do you find the domain of a composition of two functions?
2. What happens to the domain if the denominator in a rational function contains a quadratic expression?
3. How do you differentiate between the operations of function composition and function division?
4. Can the composition of two functions ever result in a restricted domain when both functions have unrestricted domains?
5. What is the importance of simplifying expressions after performing operations on functions?

**Tip:** Always check the domains carefully when performing operations like division and composition on functions, as restrictions can arise from both processes.

Given f(x) = 5x - 9 and g(x) = x - 7, find (a) (f / g)(x), Domain: and (b) (f ∘ g)(x), Domain:

Learn how to solve problems involving function division and composition with detailed solutions. Find (f / g)(x) and (f ∘ g)(x) for f(x) = 5x - 9 and g(x) = x - 7, and determine their respective domains. Suitable for students in Grades 10-12.

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