We are given two functions:

$f(x) = -3x - 1$ $g(x) = x^2 + 5$

We are asked to find $(f \circ g)(-6)$, which means evaluating the composite function $f(g(-6))$.

Step 1: Evaluate $g(-6)$

$g(x) = x^2 + 5$ Substitute $x = -6$: $g(-6) = (-6)^2 + 5 = 36 + 5 = 41$

Step 2: Evaluate $f(g(-6)) = f(41)$

$f(x) = -3x - 1$ Substitute $x = 41$: $f(41) = -3(41) - 1 = -123 - 1 = -124$

Thus, $(f \circ g)(-6) = -124$.

Let me know if you need further clarifications!

Follow-up Questions:

1. What is the general form of a composite function?
2. How would you reverse this process to solve for $(g \circ f)(-6)$?
3. How does the chain rule in calculus relate to composite functions?
4. What happens if we change the functions $f(x)$ or $g(x)$?
5. How do you find the domain of a composite function?

Tip: Always start by evaluating the innermost function first when working with composite functions.