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Which function defines [tex]\((f+g)(x)\)[/tex]?

[tex]\[
\begin{array}{l}
f(x)=\frac{5}{x}+12 \\
g(x)=\sqrt{x-3}+10
\end{array}
\][/tex]

A. [tex]\((f+g)(x)=\frac{5}{x}-\sqrt{x-3}+2\)[/tex]

B. [tex]\((f+g)(x)=\frac{\sqrt{x-3}+5}{x}+22\)[/tex]

C. [tex]\((f+g)(x)=\frac{5}{x}+\sqrt{x}+19\)[/tex]

D. [tex]\((f+g)(x)=\frac{5}{x}+\sqrt{x-3}+22\)[/tex]

Sagot :

GinnyAnswer
To determine which function defines [tex]\((f+g)(x)\)[/tex], we start by understanding the operations of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] individually.

Given:
[tex]\[ f(x) = \frac{5}{x} + 12 \][/tex]
[tex]\[ g(x) = \sqrt{x-3} + 10 \][/tex]

The composition [tex]\((f+g)(x)\)[/tex] is the sum of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:

[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]

Step-by-step:

1. Add [tex]\(f(x)\)[/tex] to [tex]\(g(x)\)[/tex]:
[tex]\[ f(x) = \frac{5}{x} + 12 \][/tex]
[tex]\[ g(x) = \sqrt{x-3} + 10 \][/tex]

So,
[tex]\[ (f+g)(x) = \left( \frac{5}{x} + 12 \right) + \left( \sqrt{x-3} + 10 \right) \][/tex]

2. Combine like terms:
[tex]\[ (f+g)(x) = \frac{5}{x} + 12 + \sqrt{x-3} + 10 \][/tex]

Simplify this further:
[tex]\[ (f+g)(x) = \frac{5}{x} + \sqrt{x-3} + 22 \][/tex]

Thus, the correct function that defines [tex]\((f+g)(x)\)[/tex] is:
[tex]\[ D. \left( f+g \right)(x) = \frac{5}{x} + \sqrt{x-3} + 22 \][/tex]
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