Let's solve the given problem step by step:

We are given the following functions:

• $f(x) = 8x + 9$
• $g(x) = 4x - 4$

(a) $f \circ g(x)$ (which is $f(g(x))$)

To find $f \circ g(x)$, we substitute $g(x) = 4x - 4$ into $f(x)$:

$f(g(x)) = f(4x - 4) = 8(4x - 4) + 9$

Simplify:

$= 8(4x) - 8(4) + 9 = 32x - 32 + 9 = 32x - 23$

So, $f \circ g(x) = 32x - 23$.

(b) $g \circ f(x)$ (which is $g(f(x))$)

To find $g \circ f(x)$, we substitute $f(x) = 8x + 9$ into $g(x)$:

$g(f(x)) = g(8x + 9) = 4(8x + 9) - 4$

Simplify:

$= 4(8x) + 4(9) - 4 = 32x + 36 - 4 = 32x + 32$

So, $g \circ f(x) = 32x + 32$.

(c) $f \circ f(x)$ (which is $f(f(x))$)

To find $f \circ f(x)$, we substitute $f(x) = 8x + 9$ into itself:

$f(f(x)) = f(8x + 9) = 8(8x + 9) + 9$

Simplify:

$= 8(8x) + 8(9) + 9 = 64x + 72 + 9 = 64x + 81$

So, $f \circ f(x) = 64x + 81$.

(d) $g \circ g(x)$ (which is $g(g(x))$)

To find $g \circ g(x)$, we substitute $g(x) = 4x - 4$ into itself:

$g(g(x)) = g(4x - 4) = 4(4x - 4) - 4$

Simplify:

$= 4(4x) - 4(4) - 4 = 16x - 16 - 4 = 16x - 20$

So, $g \circ g(x) = 16x - 20$.

Summary of Results:

• (a) $f \circ g(x) = 32x - 23$
• (b) $g \circ f(x) = 32x + 32$
• (c) $f \circ f(x) = 64x + 81$
• (d) $g \circ g(x) = 16x - 20$

Would you like further details or explanations on any step? Here are 5 additional questions that can expand your understanding:

1. How do you determine the domain of composite functions like $f \circ g(x)$?
2. What happens if one of the functions $f(x)$ or $g(x)$ is not defined at a specific value of $x$?
3. Can you give a real-life example where composite functions might be applied?
4. How would you verify if two functions are inverses of each other using composition?
5. What are the graphical interpretations of the compositions $f \circ g(x)$ and $g \circ f(x)$?

Tip: When working with function compositions, make sure to carefully substitute and simplify step by step to avoid mistakes in algebraic manipulation.