To solve the problem of finding [tex]\((f \circ g)(10)\)[/tex], where [tex]\(f(x) = x^2 + 1\)[/tex] and [tex]\(g(x) = x - 4\)[/tex], we will proceed step-by-step.
First, we need to understand what [tex]\((f \circ g)(10)\)[/tex] means. The notation [tex]\((f \circ g)(10)\)[/tex] is read as "f composed with g of 10." This means we will first apply the function [tex]\(g\)[/tex] to the number 10, and then we will apply the function [tex]\(f\)[/tex] to the result of [tex]\(g(10)\)[/tex].
1. Evaluate [tex]\(g(10)\)[/tex]:
[tex]\[
g(x) = x - 4
\][/tex]
Substituting [tex]\(x = 10\)[/tex]:
[tex]\[
g(10) = 10 - 4 = 6
\][/tex]
2. Next, evaluate [tex]\(f(g(10))\)[/tex] which is [tex]\(f(6)\)[/tex]:
[tex]\[
f(x) = x^2 + 1
\][/tex]
Substituting [tex]\(x = 6\)[/tex]:
[tex]\[
f(6) = 6^2 + 1 = 36 + 1 = 37
\][/tex]
Therefore, [tex]\((f \circ g)(10) = f(g(10)) = f(6) = 37\)[/tex].
The value equivalent to [tex]\((f \circ g)(10)\)[/tex] is [tex]\(\boxed{37}\)[/tex].
