Let's break down the solution step by step for each part.
### Part (a) [tex]\((f \circ g)(3)\)[/tex]:
This notation means we first apply the function [tex]\( g \)[/tex] to [tex]\( 3 \)[/tex], and then apply the function [tex]\( f \)[/tex] to the result of [tex]\( g(3) \)[/tex].
1. Evaluate [tex]\( g(3) \)[/tex]:
According to the definition of [tex]\( g(x) \)[/tex], for [tex]\( 3 \leq x < 9 \)[/tex], [tex]\( g(x) = -4 \)[/tex].
Therefore,
[tex]\[
g(3) = -4
\][/tex]
2. Apply [tex]\( f \)[/tex] to [tex]\( g(3) \)[/tex] (which is -4):
We now need to find [tex]\( f(-4) \)[/tex]:
[tex]\[
f(x) = 4x^2 + 5x + 1
\][/tex]
Substitute [tex]\( x = -4 \)[/tex]:
[tex]\[
f(-4) = 4(-4)^2 + 5(-4) + 1 = 4 \cdot 16 - 20 + 1 = 64 - 20 + 1 = 45
\][/tex]
So, [tex]\((f \circ g)(3) = f(g(3)) = f(-4) = 45\)[/tex].
### Part (b) [tex]\((g \circ f)(0)\)[/tex]:
This notation means we first apply the function [tex]\( f \)[/tex] to [tex]\( 0 \)[/tex], and then apply the function [tex]\( g \)[/tex] to the result of [tex]\( f(0) \)[/tex].
1. Evaluate [tex]\( f(0) \)[/tex]:
[tex]\[
f(x) = 4x^2 + 5x + 1
\][/tex]
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 4(0)^2 + 5(0) + 1 = 1
\][/tex]
2. Apply [tex]\( g \)[/tex] to [tex]\( f(0) \)[/tex] (which is 1):
According to the definition of [tex]\( g(x) \)[/tex], for [tex]\( x < 3 \)[/tex], [tex]\( g(x) = -5x + 2 \)[/tex].
Since [tex]\( 1 < 3 \)[/tex], we use the first case:
[tex]\[
g(1) = -5(1) + 2 = -5 + 2 = -3
\][/tex]
So, [tex]\((g \circ f)(0) = g(f(0)) = g(1) = -3\)[/tex].
### Final Answers:
(a) [tex]\((f \circ g)(3) = 45\)[/tex]
(b) [tex]\((g \circ f)(0) = -3\)[/tex]
