Let's go step by step through the problems:

1. Find and simplify $f(g(6))$:

We are given:

• $f(x) = x - 9$
• $g(x) = -5x - 3$

First, we need to find $g(6)$: $g(6) = -5(6) - 3 = -30 - 3 = -33$ Now, substitute $g(6) = -33$ into $f(x)$: $f(g(6)) = f(-33) = -33 - 9 = -42$ So, $f(g(6)) = -42$.

2. Find and simplify $f(g(x))$:

We are looking for $f(g(x))$, which means we substitute $g(x)$ into $f(x)$: $f(g(x)) = f(-5x - 3) = (-5x - 3) - 9 = -5x - 12$ Thus, $f(g(x)) = -5x - 12$.

3. Find and simplify $g(f(x))$:

We need to substitute $f(x)$ into $g(x)$: $g(f(x)) = g(x - 9) = -5(x - 9) - 3 = -5x + 45 - 3 = -5x + 42$ Thus, $g(f(x)) = -5x + 42$.

4. Find and simplify $(g \circ g)(x)$:

This represents $g(g(x))$, which means substituting $g(x)$ into itself: $g(g(x)) = g(-5x - 3) = -5(-5x - 3) - 3 = 25x + 15 - 3 = 25x + 12$ Thus, $(g \circ g)(x) = 25x + 12$.

5. Evaluate $(f \circ f)(-10)$:

This represents $f(f(-10))$. First, find $f(-10)$: $f(-10) = -10 - 9 = -19$ Now, substitute $-19$ back into $f(x)$: $f(-19) = -19 - 9 = -28$ Thus, $(f \circ f)(-10) = -28$.

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Summary of Results:

1. $f(g(6)) = -42$
2. $f(g(x)) = -5x - 12$
3. $g(f(x)) = -5x + 42$
4. $(g \circ g)(x) = 25x + 12$
5. $(f \circ f)(-10) = -28$

Let me know if you'd like any further details or clarifications. Here are 5 related questions:

1. What is $f(f(x))$ in general form?
2. How would you find $g(g(2))$?
3. Can you solve $f(x) = g(x)$ for $x$?
4. What is the inverse of $f(x)$, if it exists?
5. How do you find $g^{-1}(x)$ for the given $g(x)$?

Tip: When dealing with function composition, always substitute the inner function's result into the outer function step by step to avoid mistakes.