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Let's solve the function compositions step by step, given the functions:

- \( f(x) = -8x - 10 \)
- \( g(x) = -3x - 9 \)

### 1. \( (f \circ f)(x) \) or \( f(f(x)) \)
Substitute \( f(x) \) into itself:

\[
f(f(x)) = f(-8x - 10) = -8(-8x - 10) - 10
\]
Simplify:

\[
= 64x + 80 - 10 = 64x + 70
\]

Thus:
\[
(f \circ f)(x) = 64x + 70
\]

### 2. \( (f \circ g)(x) \) or \( f(g(x)) \)
Substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f(-3x - 9) = -8(-3x - 9) - 10
\]
Simplify:

\[
= 24x + 72 - 10 = 24x + 62
\]

Thus:
\[
(f \circ g)(x) = 24x + 62
\]

### 3. \( (g \circ f)(x) \) or \( g(f(x)) \)
Substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g(-8x - 10) = -3(-8x - 10) - 9
\]
Simplify:

\[
= 24x + 30 - 9 = 24x + 21
\]

Thus:
\[
(g \circ f)(x) = 24x + 21
\]

### 4. \( (g \circ g)(x) \) or \( g(g(x)) \)
Substitute \( g(x) \) into itself:

\[
g(g(x)) = g(-3x - 9) = -3(-3x - 9) - 9
\]
Simplify:

\[
= 9x + 27 - 9 = 9x + 18
\]

Thus:
\[
(g \circ g)(x) = 9x + 18
\]

### Final Answers:
1. \( (f \circ f)(x) = 64x + 70 \)
2. \( (f \circ g)(x) = 24x + 62 \)
3. \( (g \circ f)(x) = 24x + 21 \)
4. \( (g \circ g)(x) = 9x + 18 \)

Would you like more details on any step? Here are five related questions:

1. How do you determine the domain of composed functions?
2. What is the general rule for the composition of two functions?
3. How can you verify if two functions are inverses through composition?
4. What is the difference between \( f \circ g \) and \( g \circ f \)?
5. How does function composition relate to transformations of graphs?

**Tip**: Always check if function composition results in a simplified expression that defines the domain and range clearly.

Given f(x) = -8x - 10 and g(x) = -3x - 9, determine the compositions: (f ◦ f)(x), (f ◦ g)(x), (g ◦ f)(x), and (g ◦ g)(x).

Learn how to compute the composition of two linear functions f(x) = -8x - 10 and g(x) = -3x - 9. This problem involves finding (f ◦ f)(x), (f ◦ g)(x), (g ◦ f)(x), and (g ◦ g)(x) with detailed step-by-step solutions.

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