If [tex]\( f(x)=\frac{x+4}{x}, g(x)=x-2 \)[/tex], and [tex]\( h(x)=4x-1 \)[/tex], what is [tex]\( (f \circ h \circ g)(x) \)[/tex] ?

A. [tex]\( (f \circ h \circ g)(x)=\frac{2x+18}{x} \)[/tex]
B. [tex]\( (f \circ h \circ g)(x)=\frac{2x+4}{x} \)[/tex]
C. [tex]\( (f \circ h \circ g)(x)=\frac{4x-3}{4x-7} \)[/tex]
D. [tex]\( (f \circ h \circ g)(x)=\frac{4x-5}{4x-8} \)[/tex]



Answer :

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.