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### Given:
- \( f(x) = 2x - 3 \)
- \( g(x) = 3x + 2 \)

#### 1. Find \( f(0) \) and \( g(f(0)) \).
   - \( f(0) = 2(0) - 3 = -3 \)
   - \( g(f(0)) = g(-3) = 3(-3) + 2 = -9 + 2 = -7 \)

#### 2. Find \( (g \circ f)(x) \), and state the domain of \( g \circ f \).
   - \( (g \circ f)(x) = g(f(x)) = g(2x - 3) = 3(2x - 3) + 2 = 6x - 9 + 2 = 6x - 7 \)
   - The domain of \( g \circ f \) is all real numbers \( \mathbb{R} \), because both \( f(x) \) and \( g(x) \) are linear functions, and thus defined for all real \( x \).

#### 3. Use the result in Number 2 to evaluate \( (g \circ f)(0) \).
   - \( (g \circ f)(0) = 6(0) - 7 = -7 \)

#### 4. Find \( g(0) \) and \( f(g(0)) \).
   - \( g(0) = 3(0) + 2 = 2 \)
   - \( f(g(0)) = f(2) = 2(2) - 3 = 4 - 3 = 1 \)

#### 5. Find \( (f \circ g)(x) \), and state the domain of \( f \circ g \).
   - \( (f \circ g)(x) = f(g(x)) = f(3x + 2) = 2(3x + 2) - 3 = 6x + 4 - 3 = 6x + 1 \)
   - The domain of \( f \circ g \) is all real numbers \( \mathbb{R} \), as both functions are linear.

#### 6. Use the result in Number 5 to evaluate \( (f \circ g)(0) \).
   - \( (f \circ g)(0) = 6(0) + 1 = 1 \)

### Summary:
1. \( f(0) = -3 \), \( g(f(0)) = -7 \)
2. \( (g \circ f)(x) = 6x - 7 \), Domain: \( \mathbb{R} \)
3. \( (g \circ f)(0) = -7 \)
4. \( g(0) = 2 \), \( f(g(0)) = 1 \)
5. \( (f \circ g)(x) = 6x + 1 \), Domain: \( \mathbb{R} \)
6. \( (f \circ g)(0) = 1 \)

Would you like further details or have any questions? Here are five related questions for deeper understanding:

1. How do you find the inverse of a function?
2. What is the difference between the domain and range of a function?
3. How do you determine if a function is one-to-one?
4. What are the steps to combine functions through addition or multiplication?
5. How can you determine the composite function \( h \circ g \circ f(x) \)?

**Tip:** Always check the domain of each function before determining the domain of a composite function, as the domain of the composite function is restricted to the domain where both functions are defined.

Let f(x) = 2x - 3 and g(x) = 3x + 2.
1. Find f(0) and g(f(0)).
2. Find (g ∘ f)(x), and state the domain of g ∘ f.
3. Use the result in Number 2 to evaluate (g ∘ f)(0).
4. Find g(0) and f(g(0)).
5. Find (f ∘ g)(x), and state the domain of f ∘ g.
6. Use the result in Number 5 to evaluate (f ∘ g)(0).

Learn how to solve problems involving function composition with linear functions. This guide includes detailed steps to find (g ∘ f)(x), (f ∘ g)(x), and their domains, along with evaluations at specific points. Suitable for Grades 9-10 students.

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