Let’s break this problem into parts to solve the expressions for each question.

We are given:

• $f(x) = 9x$
• $g(x) = 5x^2 + 8$

The questions involve composite functions:

(a) $(f \circ g)(4) = f(g(4))$

1. Calculate $g(4)$: $g(4) = 5(4)^2 + 8 = 5(16) + 8 = 80 + 8 = 88$
2. Use $g(4) = 88$ to find $f(g(4))$: $f(88) = 9(88) = 792$ Answer: $(f \circ g)(4) = 792$

(b) $(g \circ f)(2) = g(f(2))$

1. Calculate $f(2)$: $f(2) = 9(2) = 18$
2. Use $f(2) = 18$ to find $g(f(2))$: $g(18) = 5(18)^2 + 8 = 5(324) + 8 = 1620 + 8 = 1628$ Answer: $(g \circ f)(2) = 1628$

(c) $(f \circ f)(1) = f(f(1))$

1. Calculate $f(1)$: $f(1) = 9(1) = 9$
2. Use $f(1) = 9$ to find $f(f(1))$: $f(9) = 9(9) = 81$ Answer: $(f \circ f)(1) = 81$

(d) $(g \circ g)(0) = g(g(0))$

1. Calculate $g(0)$: $g(0) = 5(0)^2 + 8 = 0 + 8 = 8$
2. Use $g(0) = 8$ to find $g(g(0))$: $g(8) = 5(8)^2 + 8 = 5(64) + 8 = 320 + 8 = 328$ Answer: $(g \circ g)(0) = 328$

Final Answers:

(a) $792$
(b) $1628$
(c) $81$
(d) $328$

Would you like a detailed explanation of any specific step? Here are some follow-up questions you might consider:

1. What are composite functions and how do they work?
2. Can you explain how the individual calculations were derived?
3. How do we simplify expressions with powers and coefficients?
4. How do these concepts apply to real-world mathematical problems?
5. What happens if $f(x)$ or $g(x)$ changes?

Tip: Always calculate the inner function first when working with composite functions.