To solve the problem and find [tex]\((f \circ h \circ g)(x)\)[/tex], let's break it down step-by-step. We will need to compute [tex]\(g(x)\)[/tex], then [tex]\(h(g(x))\)[/tex], and finally [tex]\(f(h(g(x)))\)[/tex].
1. First, compute [tex]\(g(x)\)[/tex]:
[tex]\[
g(x) = x - 2
\][/tex]
2. Next, use the result of [tex]\(g(x)\)[/tex] in [tex]\(h(x)\)[/tex]:
[tex]\[
h(g(x)) = h(x - 2)
\][/tex]
Substitute [tex]\(x - 2\)[/tex] into the function [tex]\(h(x) = 4x - 1\)[/tex]:
[tex]\[
h(x - 2) = 4(x - 2) - 1 = 4x - 8 - 1 = 4x - 9
\][/tex]
So, [tex]\(h(g(x)) = 4x - 9\)[/tex].
3. Finally, apply the result of [tex]\(h(g(x))\)[/tex] to [tex]\(f(x)\)[/tex]:
[tex]\[
f(h(g(x))) = f(4x - 9)
\][/tex]
Substitute [tex]\(4x - 9\)[/tex] into the function [tex]\(f(x) = \frac{x + 4}{x}\)[/tex]:
[tex]\[
f(4x - 9) = \frac{(4x - 9) + 4}{4x - 9} = \frac{4x - 5}{4x - 9}
\][/tex]
Therefore, the composition [tex]\((f \circ h \circ g)(x)\)[/tex] is:
[tex]\[
(f \circ h \circ g)(x) = \frac{4x - 5}{4x - 9}
\][/tex]
Let's match our result with the given options:
- Option 1: [tex]\(\frac{2x + 18}{x}\)[/tex]
- Option 2: [tex]\(\frac{2x + 4}{x}\)[/tex]
- Option 3: [tex]\(\frac{4x - 3}{4x - 7}\)[/tex]
- Option 4: [tex]\(\frac{4x - 5}{4x - 9}\)[/tex]
The correct answer is:
[tex]\[
(f \circ h \circ g)(x) = \frac{4x - 5}{4x - 9}
\][/tex]
which is Option 4.
