To find [tex]\((g \circ h \circ f)(x)\)[/tex], we need to perform the composition of the functions [tex]\(f\)[/tex], [tex]\(h\)[/tex], and [tex]\(g\)[/tex] in that order. This means we apply [tex]\(f\)[/tex] first, then [tex]\(h\)[/tex], and finally [tex]\(g\)[/tex].
1. Compute [tex]\(f(x)\)[/tex]:
[tex]\[
f(x) = \frac{x - 3}{x}
\][/tex]
2. Compute [tex]\(h(f(x))\)[/tex]:
Substitute [tex]\(f(x)\)[/tex] into [tex]\(h\)[/tex]:
[tex]\[
h(f(x)) = h\left(\frac{x - 3}{x}\right)
\][/tex]
Given [tex]\(h(t) = 2t + 1\)[/tex], where [tex]\(t\)[/tex] is the input,
[tex]\[
h\left(\frac{x - 3}{x}\right) = 2 \left(\frac{x - 3}{x}\right) + 1
\][/tex]
Distribute the 2:
[tex]\[
= \frac{2(x - 3)}{x} + 1
\][/tex]
Combine the terms over a common denominator:
[tex]\[
= \frac{2(x - 3) + x}{x} = \frac{2x - 6 + x}{x} = \frac{3x - 6}{x}
\][/tex]
3. Compute [tex]\(g(h(f(x)))\)[/tex]:
Substitute [tex]\(h(f(x))\)[/tex] into [tex]\(g\)[/tex]:
[tex]\[
g\left(h(f(x))\right) = g\left(\frac{3x - 6}{x}\right)
\][/tex]
Given [tex]\(g(t) = t + 3\)[/tex],
[tex]\[
g\left(\frac{3x - 6}{x}\right) = \left(\frac{3x - 6}{x}\right) + 3
\][/tex]
Express [tex]\(3\)[/tex] with a common denominator of [tex]\(x\)[/tex]:
[tex]\[
= \frac{3x - 6}{x} + \frac{3x}{x}
\][/tex]
Combine the terms:
[tex]\[
= \frac{3x - 6 + 3x}{x} = \frac{6x - 6}{x}
\][/tex]
So, [tex]\((g \circ h \circ f)(x) = \frac{6x - 6}{x}\)[/tex].
This corresponds to the second option:
[tex]\[
(g \circ h \circ f)(x) = \frac{6 x-6}{x}
\][/tex]
