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The problem asks to find \( g(x) \) given the following information:

1. \( f(x) = 2x + 3 \)
2. \( (f + g)(x) = 3x - 5 \)

### Steps to solve for \( g(x) \):

- By definition, \( (f + g)(x) = f(x) + g(x) \).
- This means that:
  \[
  f(x) + g(x) = 3x - 5
  \]
- Substituting \( f(x) = 2x + 3 \) into the equation:
  \[
  (2x + 3) + g(x) = 3x - 5
  \]
- Solving for \( g(x) \):
  \[
  g(x) = (3x - 5) - (2x + 3)
  \]
  \[
  g(x) = 3x - 5 - 2x - 3
  \]
  \[
  g(x) = x - 8
  \]

Thus, \( g(x) = x - 8 \).

Would you like a more detailed explanation, or do you have any questions? 

Here are 5 related questions to deepen your understanding:
1. How would you solve for \( g(x) \) if \( f(x) \) had a quadratic form instead of linear?
2. What is the domain of \( f(x) \) and \( g(x) \) in this case?
3. Can you represent the sum of two functions graphically and how would this apply to \( f(x) + g(x) \)?
4. What happens to the combined function \( f(x) + g(x) \) if you subtract a constant from both functions?
5. How would this change if you were given \( (f \cdot g)(x) \) instead of \( (f + g)(x) \)?

**Tip:** When solving for a function in terms of others, carefully substitute given expressions and simplify step-by-step.

If f(x) = 2x + 3 and (f + g)(x) = 3x - 5, find g(x).

Learn how to find g(x) given f(x) = 2x + 3 and (f + g)(x) = 3x - 5. This problem involves key algebraic concepts like function addition and step-by-step substitution. Suitable for Grades 7-9.

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