To solve the problem, let us analyze the given functions and evaluate each composite expression step by step.

Given:

• $f(x) = 9x$
• $g(x) = 7x^2 + 4$

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

This means $f(g(4))$. First, calculate $g(4)$: $g(4) = 7(4^2) + 4 = 7(16) + 4 = 112 + 4 = 116$ Now calculate $f(116)$: $f(116) = 9(116) = 1044$ Thus, $(f \circ g)(4) = 1044$.

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

This means $g(f(2))$. First, calculate $f(2)$: $f(2) = 9(2) = 18$ Now calculate $g(18)$: $g(18) = 7(18^2) + 4 = 7(324) + 4 = 2268 + 4 = 2272$ Thus, $(g \circ f)(2) = 2272$.

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

This means $f(f(1))$. First, calculate $f(1)$: $f(1) = 9(1) = 9$ Now calculate $f(9)$: $f(9) = 9(9) = 81$ Thus, $(f \circ f)(1) = 81$.

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

This means $g(g(0))$. First, calculate $g(0)$: $g(0) = 7(0^2) + 4 = 4$ Now calculate $g(4)$: $g(4) = 7(4^2) + 4 = 7(16) + 4 = 112 + 4 = 116$ Thus, $(g \circ g)(0) = 116$.

Final Results:

• (a) $(f \circ g)(4) = 1044$
• (b) $(g \circ f)(2) = 2272$
• (c) $(f \circ f)(1) = 81$
• (d) $(g \circ g)(0) = 116$

Let me know if you need a detailed explanation or help with related questions!

Relative Questions:

1. What is the general formula for evaluating composite functions?
2. How can we simplify the evaluation of quadratic functions like $g(x) = 7x^2 + 4$?
3. What happens if $f(x)$ or $g(x)$ is not defined for certain values of $x$?
4. Can composite functions result in undefined values? When does this occur?
5. How can we visualize the behavior of $f(x)$ and $g(x)$ on a graph?

Tip: Always calculate inner functions first when working with compositions to avoid mistakes!