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We are given two functions:

- \( f(x) = 5x + 6 \)
- \( g(x) = -3 \)

The questions are:

(a) \( f(g(4)) \)
(b) \( g(f(-5)) \)

Let's solve them step by step.

### (a) \( f(g(4)) \)

1. First, we find \( g(4) \). Since \( g(x) = -3 \) for all \( x \), we know that:
   \[
   g(4) = -3
   \]
2. Next, we substitute \( g(4) = -3 \) into \( f(x) \):
   \[
   f(-3) = 5(-3) + 6 = -15 + 6 = -9
   \]

Thus, \( f(g(4)) = -9 \).

### (b) \( g(f(-5)) \)

1. First, we find \( f(-5) \):
   \[
   f(-5) = 5(-5) + 6 = -25 + 6 = -19
   \]
2. Next, we substitute \( f(-5) = -19 \) into \( g(x) \). Since \( g(x) = -3 \) for all \( x \), we know that:
   \[
   g(-19) = -3
   \]

Thus, \( g(f(-5)) = -3 \).

### Final Answers:
- \( f(g(4)) = -9 \)
- \( g(f(-5)) = -3 \)

Would you like further details or have any questions about the steps?

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Here are some related questions:
1. What is the general definition of composite functions?
2. How would you approach calculating \( f(g(x)) \) for a new pair of functions?
3. Can a constant function like \( g(x) = -3 \) ever be injective or surjective?
4. What happens if \( f(x) \) and \( g(x) \) are both linear functions?
5. How would you graph \( f(x) = 5x + 6 \) and \( g(x) = -3 \)?

**Tip:** Always start solving composite functions from the inside and work your way outward!

Given that f(x) = 5x + 6 and g(x) = -3, calculate f(g(4)) and g(f(-5))

Learn how to solve composite functions like f(g(x)) and g(f(x)) with detailed steps. In this problem, f(x) is a linear function and g(x) is a constant function, providing a clear example of function composition. Suitable for students in Grades 9-12.

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