To evaluate the function [tex]\((g \circ f)(x)\)[/tex] for [tex]\(x=2\)[/tex], we need to follow these steps:
1. Evaluate [tex]\(f(2)\)[/tex]:
The function [tex]\(f(x)\)[/tex] is defined as:
[tex]\[
f(x) = x^3 + 3x
\][/tex]
Substituting [tex]\(x = 2\)[/tex]:
[tex]\[
f(2) = 2^3 + 3 \cdot 2 = 8 + 6 = 14
\][/tex]
2. Evaluate [tex]\(g(f(2))\)[/tex]:
From the first step, we found that [tex]\(f(2) = 14\)[/tex]. Now, we need to find [tex]\(g(14)\)[/tex].
The function [tex]\(g(x)\)[/tex] is defined as:
[tex]\[
g(x) = \sqrt{5x}
\][/tex]
Substituting [tex]\(x = 14\)[/tex]:
[tex]\[
g(14) = \sqrt{5 \cdot 14} = \sqrt{70}
\][/tex]
3. Simplify the expression:
The simplest exact form for [tex]\(g(14)\)[/tex] is [tex]\(\sqrt{70}\)[/tex]. Converting this to a numerical value:
[tex]\[
\sqrt{70} \approx 8.366600265340756
\][/tex]
Putting all this together, we conclude that:
[tex]\[
(g \circ f)(2) = g(f(2)) = g(14) = \sqrt{70}
\][/tex]
Therefore, [tex]\((g \circ f)(2)\)[/tex] is:
[tex]\[
(g \circ f)(2) = \sqrt{70} \approx 8.366600265340756
\][/tex]
The boxed answer is:
[tex]\[
(g \circ f)(2) = \sqrt{70}
\][/tex]
