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.