Sure, let's work through the problem step-by-step, considering the functions [tex]\( f(x) = 8x \)[/tex] and [tex]\( g(x) = x + 8 \)[/tex].
### a. [tex]\((f \circ g)(x)\)[/tex]
This notation [tex]\((f \circ g)(x)\)[/tex] represents the composition of the functions [tex]\(f\)[/tex] and [tex]\(g\)[/tex], which means we apply [tex]\(g(x)\)[/tex] first, and then apply [tex]\(f\)[/tex] to the result of [tex]\(g(x)\)[/tex].
1. First, compute [tex]\(g(x)\)[/tex]:
[tex]\[
g(x) = x + 8
\][/tex]
2. Next, substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[
f(g(x)) = f(x + 8)
\][/tex]
3. Now, apply the function [tex]\(f\)[/tex] to [tex]\(x + 8\)[/tex]:
[tex]\[
f(x + 8) = 8(x + 8)
\][/tex]
4. Simplify:
[tex]\[
8(x + 8) = 8x + 64
\][/tex]
Therefore,
[tex]\[
(f \circ g)(x) = 8x + 64
\][/tex]
### b. [tex]\((g \circ f)(x)\)[/tex]
This notation [tex]\((g \circ f)(x)\)[/tex] represents the composition of the functions [tex]\(g\)[/tex] and [tex]\(f\)[/tex], which means we apply [tex]\(f(x)\)[/tex] first, and then apply [tex]\(g\)[/tex] to the result of [tex]\(f(x)\)[/tex].
1. First, compute [tex]\(f(x)\)[/tex]:
[tex]\[
f(x) = 8x
\][/tex]
2. Next, substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[
g(f(x)) = g(8x)
\][/tex]
3. Now, apply the function [tex]\(g\)[/tex] to [tex]\(8x\)[/tex]:
[tex]\[
g(8x) = 8x + 8
\][/tex]
Therefore,
[tex]\[
(g \circ f)(x) = 8x + 8
\][/tex]
### c. [tex]\((f \circ g)(2)\)[/tex]
To find [tex]\((f \circ g)(2)\)[/tex], we'll use the function we found in part (a).
1. Substitute [tex]\(x = 2\)[/tex] into [tex]\((f \circ g)(x)\)[/tex]:
[tex]\[
(f \circ g)(2) = 8(2) + 64
\][/tex]
2. Simplify:
[tex]\[
16 + 64 = 80
\][/tex]
Therefore,
[tex]\[
(f \circ g)(2) = 80
\][/tex]
### Summary of Results:
a. [tex]\((f \circ g)(x) = 8x + 64\)[/tex]
b. [tex]\((g \circ f)(x) = 8x + 8\)[/tex]
c. [tex]\((f \circ g)(2) = 80\)[/tex]
Therefore, when evaluated, the results are:
[tex]\[
((f \circ g)(2), (g \circ f)(2), (f \circ g)(2)) = (80, 24, 80)
\][/tex]
