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Let's break down the functions and solve each part of the question step by step.

We are given three functions:

\[
f(x) = 4 - 2x
\]
\[
g(x) = x^2 + 5
\]
\[
h(x) = \frac{x - 6}{7}
\]

### Part a: \( f(g(-3)) \)

We are told \( f(g(-3)) = -24 \), but let's confirm this.

1. First, find \( g(-3) \):
   \[
   g(x) = x^2 + 5
   \]
   Substituting \( x = -3 \):
   \[
   g(-3) = (-3)^2 + 5 = 9 + 5 = 14
   \]

2. Now, find \( f(14) \):
   \[
   f(x) = 4 - 2x
   \]
   Substituting \( x = 14 \):
   \[
   f(14) = 4 - 2(14) = 4 - 28 = -24
   \]

Thus, \( f(g(-3)) = -24 \), which is correctly given.

### Part b: \( g \circ h(20) \) (This means \( g(h(20)) \))

1. First, find \( h(20) \):
   \[
   h(x) = \frac{x - 6}{7}
   \]
   Substituting \( x = 20 \):
   \[
   h(20) = \frac{20 - 6}{7} = \frac{14}{7} = 2
   \]

2. Now, find \( g(2) \):
   \[
   g(x) = x^2 + 5
   \]
   Substituting \( x = 2 \):
   \[
   g(2) = 2^2 + 5 = 4 + 5 = 9
   \]

Thus, \( g \circ h(20) = 9 \).

### Part c: \( h(f(g(3))) \)

1. First, find \( g(3) \):
   \[
   g(3) = 3^2 + 5 = 9 + 5 = 14
   \]

2. Now, find \( f(14) \) (which we already computed in Part a):
   \[
   f(14) = -24
   \]

3. Now, find \( h(-24) \):
   \[
   h(x) = \frac{x - 6}{7}
   \]
   Substituting \( x = -24 \):
   \[
   h(-24) = \frac{-24 - 6}{7} = \frac{-30}{7}
   \]

Thus, \( h(f(g(3))) = \frac{-30}{7} \).

### Final Answers:
a. \( f(g(-3)) = -24 \) (Given)
b. \( g \circ h(20) = 9 \)
c. \( h(f(g(3))) = \frac{-30}{7} \)

Would you like further details or explanations on any part of this?

Here are some related questions:
1. How would you compute \( f(h(g(x))) \) for an arbitrary \( x \)?
2. What would \( h(f(h(x))) \) be for a general input \( x \)?
3. Can you find the composite function \( f \circ g(x) \) for a general \( x \)?
4. What would the graph of \( h(x) \) look like, and how would its slope affect the composite functions?
5. If \( g(x) \) were different, say \( g(x) = x^2 - 5 \), how would that change the result for part c?

**Tip:** When dealing with composite functions, always evaluate the innermost function first before moving to the outer ones.

Determine the following based on the functions: f(x) = 4 - 2x, g(x) = x^2 + 5, h(x) = (x - 6) / 7. a) f(g(-3)) = -24 b) g ∘ h(20) c) h(f(g(3))).

Learn how to evaluate composite functions using f(x) = 4 - 2x, g(x) = x^2 + 5, and h(x) = (x - 6) / 7. This example covers function composition with detailed steps and solutions for a high school-level algebra problem.

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