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Let's break down the problem.

You are given two functions:

- \( f(x) = x^2 + 9x \)
- \( g(x) = x - 7 \)

You are asked to evaluate two function compositions:

### (a) \( (f \circ g)(5) \)
This is equivalent to \( f(g(5)) \). First, we will evaluate \( g(5) \), then substitute the result into \( f(x) \).

1. \( g(5) = 5 - 7 = -2 \)
2. Now, evaluate \( f(-2) \):
   \[
   f(-2) = (-2)^2 + 9(-2) = 4 - 18 = -14
   \]
Thus, \( (f \circ g)(5) = -14 \).

### (b) \( (g \circ f)(5) \)
This is equivalent to \( g(f(5)) \). First, we will evaluate \( f(5) \), then substitute the result into \( g(x) \).

1. \( f(5) = (5)^2 + 9(5) = 25 + 45 = 70 \)
2. Now, evaluate \( g(70) \):
   \[
   g(70) = 70 - 7 = 63
   \]
Thus, \( (g \circ f)(5) = 63 \).

Both answers match the given values in the problem:

- \( (f \circ g)(5) = -14 \)
- \( (g \circ f)(5) = 63 \)

Would you like further explanation on function composition or any additional details?

Here are 5 follow-up questions to explore this topic further:

1. What is the general rule for function composition?
2. How would you find \( (f \circ g)(x) \) for a general \( x \)?
3. How does the order of composition affect the result? Why is \( f \circ g \) different from \( g \circ f \)?
4. Can you find \( (f \circ f)(2) \) and \( (g \circ g)(3) \) for extra practice?
5. How do you interpret compositions of functions graphically?

**Tip:** In function composition, always apply the inner function first, then use the result to evaluate the outer function.

If f(x) = x^2 + 9x and g(x) = x - 7, evaluate each of the following: (a) (f ∘ g)(5), (b) (g ∘ f)(5).

Evaluate the function compositions (f ∘ g)(5) and (g ∘ f)(5) given f(x) = x^2 + 9x and g(x) = x - 7. This step-by-step solution explores key concepts in function composition and algebra, suitable for students in Grades 9-10.

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