To find the correct function that defines [tex]\((f+g)(x)\)[/tex], we start by analyzing the given functions and then combining them.
Given functions:
[tex]\[ f(x) = \frac{5}{x} + 12 \][/tex]
[tex]\[ g(x) = \sqrt{x-3} + 10 \][/tex]
Step-by-step process:
1. Compute [tex]\((f+g)(x)\)[/tex]:
To combine [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] into [tex]\((f+g)(x)\)[/tex], we sum the two functions:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Substituting the given definitions of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f+g)(x) = \left(\frac{5}{x} + 12\right) + \left(\sqrt{x-3} + 10\right) \][/tex]
2. Simplify the expression:
Combine the terms:
[tex]\[ (f+g)(x) = \frac{5}{x} + 12 + \sqrt{x-3} + 10 \][/tex]
[tex]\[ (f+g)(x) = \frac{5}{x} + \sqrt{x-3} + 22 \][/tex]
3. Verification through substitution:
To ensure the resultant function is correct, we can test for specific [tex]\(x\)[/tex] values and match these calculations with the given options.
4. Choosing [tex]\(x = 4\)[/tex] as a sample value:
- Calculate [tex]\(f(4)\)[/tex]:
[tex]\[ f(4) = \frac{5}{4} + 12 = 1.25 + 12 = 13.25 \][/tex]
- Calculate [tex]\(g(4)\)[/tex]:
[tex]\[ g(4) = \sqrt{4-3} + 10 = \sqrt{1} + 10 = 1 + 10 = 11.00 \][/tex]
- Therefore, [tex]\((f+g)(4)\)[/tex]:
[tex]\[ (f+g)(4) = 13.25 + 11.00 = 24.25 \][/tex]
5. Compare with the options:
Options given:
- A. [tex]\(\frac{5}{x} + \sqrt{x} + 19\)[/tex]
- B. [tex]\(\frac{5}{x} - \sqrt{x-3} + 2\)[/tex]
- C. [tex]\(\frac{\sqrt{2-3} + 5}{x} + 22\)[/tex]
- D. [tex]\(\frac{5}{x} + \sqrt{x-3} + 22\)[/tex]
6. Verification:
From our combined and simplified expression for [tex]\((f+g)(x)\)[/tex], we observe it matches option D:
[tex]\[ (f+g)(x) = \frac{5}{x} + \sqrt{x-3} + 22 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{D. \left( \frac{5}{x} + \sqrt{x-3} + 22 \right)} \][/tex]
