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We are asked to find the value of \( (f \circ g)(5) \), which represents the composition of the functions \( f(x) \) and \( g(x) \).

### Given functions:
- \( f(x) = 3x^2 + 6x + 15 \)
- \( g(x) = -x + 1 \)

### Composition process:
1. First, we calculate \( g(5) \), because the composition \( (f \circ g)(5) \) means we apply \( g(x) \) first:
   \[
   g(5) = -(5) + 1 = -5 + 1 = -4
   \]

2. Next, we substitute \( g(5) = -4 \) into \( f(x) \) to find \( f(g(5)) \), which is \( f(-4) \):
   \[
   f(-4) = 3(-4)^2 + 6(-4) + 15
   \]
   \[
   f(-4) = 3(16) + 6(-4) + 15 = 48 - 24 + 15 = 39
   \]

### Final result:
\[
(f \circ g)(5) = 39
\]

Would you like more details, or do you have any questions?

Here are 5 related questions to consider:
1. How do you compute the composition of two functions in general?
2. What happens when you compose functions in the opposite order, \( (g \circ f)(x) \)?
3. How does the function composition affect the domain of the composed function?
4. What is the geometric interpretation of composing two functions?
5. How would you find the derivative of a composed function \( (f \circ g)(x) \)?

**Tip:** Always apply the inner function first when dealing with function composition, and substitute its result into the outer function.

Given the definitions of f(x) = 3x^2 + 6x + 15 and g(x) = -x + 1, find the value of (f ∘ g)(5).

Learn how to solve a function composition problem by finding (f ∘ g)(5), where f(x) is a quadratic function and g(x) is a linear function. Detailed step-by-step solution provided. Suitable for students in Grades 9-11.

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