Sure, let's go through the detailed steps to evaluate the given expressions using the functions [tex]\( f(x) = 6x^2 \)[/tex] and [tex]\( g(x) = \frac{x-8}{2} \)[/tex].
### Part (a) Evaluate [tex]\((f \circ g)(2)\)[/tex]
To find [tex]\((f \circ g)(2)\)[/tex], we need to evaluate [tex]\( f \)[/tex] at [tex]\( g(2) \)[/tex]. This involves two steps:
1. Evaluate [tex]\( g(2) \)[/tex]:
[tex]\[
g(x) = \frac{x - 8}{2}
\][/tex]
Substituting [tex]\( x = 2 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(2) = \frac{2 - 8}{2} = \frac{-6}{2} = -3
\][/tex]
2. Evaluate [tex]\( f \)[/tex] at [tex]\( g(2) \)[/tex], which is now [tex]\( f(-3) \)[/tex]:
[tex]\[
f(x) = 6x^2
\][/tex]
Substituting [tex]\( x = -3 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(-3) = 6(-3)^2 = 6 \times 9 = 54
\][/tex]
Therefore, [tex]\((f \circ g)(2) = 54\)[/tex].
### Part (b) Evaluate [tex]\((g \circ f)(-6)\)[/tex]
To find [tex]\((g \circ f)(-6)\)[/tex], we need to evaluate [tex]\( g \)[/tex] at [tex]\( f(-6) \)[/tex]. This involves two steps:
1. Evaluate [tex]\( f(-6) \)[/tex]:
[tex]\[
f(x) = 6x^2
\][/tex]
Substituting [tex]\( x = -6 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(-6) = 6(-6)^2 = 6 \times 36 = 216
\][/tex]
2. Evaluate [tex]\( g \)[/tex] at [tex]\( f(-6) \)[/tex], which is now [tex]\( g(216) \)[/tex]:
[tex]\[
g(x) = \frac{x - 8}{2}
\][/tex]
Substituting [tex]\( x = 216 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(216) = \frac{216 - 8}{2} = \frac{208}{2} = 104
\][/tex]
Therefore, [tex]\((g \circ f)(-6) = 104\)[/tex].
### Summary:
- [tex]\((f \circ g)(2) = 54\)[/tex]
- [tex]\((g \circ f)(-6) = 104\)[/tex]
