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We are given two functions \( f(x) = 1 - 3x \) and \( g(x) = \cos(x) \). We need to find the composite functions \( (f \circ g)(x) \), \( (g \circ f)(x) \), \( (f \circ f)(x) \), and \( (g \circ g)(x) \), and their domains.

### (a) \( f \circ g \):
This represents the composition \( f(g(x)) \).

\[
f(g(x)) = f(\cos(x)) = 1 - 3\cos(x)
\]

**Domain**: The domain of \( g(x) = \cos(x) \) is \( (-\infty, \infty) \), and \( f(x) = 1 - 3x \) is defined for all real numbers. Therefore, the domain of \( f(g(x)) \) is \( (-\infty, \infty) \).

### (b) \( g \circ f \):
This represents the composition \( g(f(x)) \).

\[
g(f(x)) = g(1 - 3x) = \cos(1 - 3x)
\]

**Domain**: The cosine function is defined for all real numbers, and since \( f(x) = 1 - 3x \) is also defined for all \( x \), the domain of \( g(f(x)) \) is \( (-\infty, \infty) \).

### (c) \( f \circ f \):
This represents the composition \( f(f(x)) \).

\[
f(f(x)) = f(1 - 3x) = 1 - 3(1 - 3x) = 1 - 3 + 9x = -2 + 9x
\]

**Domain**: Since \( f(x) \) is defined for all real \( x \), the domain of \( f(f(x)) \) is \( (-\infty, \infty) \).

### (d) \( g \circ g \):
This represents the composition \( g(g(x)) \).

\[
g(g(x)) = g(\cos(x)) = \cos(\cos(x))
\]

**Domain**: Since both \( g(x) = \cos(x) \) and \( \cos(x) \) are defined for all real \( x \), the domain of \( g(g(x)) \) is \( (-\infty, \infty) \).

### Summary of Results:
- \( f \circ g(x) = 1 - 3\cos(x) \), domain: \( (-\infty, \infty) \)
- \( g \circ f(x) = \cos(1 - 3x) \), domain: \( (-\infty, \infty) \)
- \( f \circ f(x) = -2 + 9x \), domain: \( (-\infty, \infty) \)
- \( g \circ g(x) = \cos(\cos(x)) \), domain: \( (-\infty, \infty) \)

Let me know if you need more details or clarifications!

Here are some follow-up questions you could explore:
1. What is the range of \( f(x) \) and \( g(x) \)?
2. How do composite functions behave with trigonometric and linear functions?
3. Can you graph \( f \circ g(x) \) and \( g \circ f(x) \)?
4. How would \( f \circ g(x) \) and \( g \circ f(x) \) behave for specific values of \( x \)?
5. How do the domains of more complex composite functions change with different inner functions?

**Tip**: When determining the domain of a composite function, always consider the domain of the inner function and how it affects the outer function's domain.

Find the functions (a) f ∘ g, (b) g ∘ f, (c) f ∘ f, and (d) g ∘ g and their domains. Given f(x) = 1 - 3x and g(x) = cos(x).

Learn how to find the composite functions f ∘ g, g ∘ f, f ∘ f, and g ∘ g, along with their domains. This problem involves linear and trigonometric functions and covers important concepts like function composition. Suitable for students in Grades 11-12.

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