Sure, let's go through the calculation step-by-step for the given functions [tex]\( f(x) = 6x + 3 \)[/tex] and [tex]\( g(x) = 3 - x^2 \)[/tex].
### Part (a)
We need to find [tex]\( f(g(0)) \)[/tex].
1. First, calculate [tex]\( g(0) \)[/tex]:
[tex]\[
g(0) = 3 - (0)^2 = 3
\][/tex]
2. Next, evaluate [tex]\( f \)[/tex] at [tex]\( g(0) \)[/tex] which is [tex]\( f(3) \)[/tex]:
[tex]\[
f(3) = 6 \cdot 3 + 3 = 18 + 3 = 21
\][/tex]
So, [tex]\( f(g(0)) = 21 \)[/tex].
### Part (b)
We need to find [tex]\( g(f(0)) \)[/tex].
1. First, calculate [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = 6 \cdot 0 + 3 = 3
\][/tex]
2. Next, evaluate [tex]\( g \)[/tex] at [tex]\( f(0) \)[/tex] which is [tex]\( g(3) \)[/tex]:
[tex]\[
g(3) = 3 - (3)^2 = 3 - 9 = -6
\][/tex]
So, [tex]\( g(f(0)) = -6 \)[/tex].
### Summary:
(a) [tex]\( f(g(0)) = 21 \)[/tex]
(b) [tex]\( g(f(0)) = -6 \)[/tex]
