To find the value of [tex]\((f \circ g)(10)\)[/tex], we need to evaluate the composition of the two functions given, which is [tex]\(f(g(10))\)[/tex]. Let's break this down step-by-step:
1. Understanding the functions:
- [tex]\(f(x) = x^2 + 1\)[/tex]
- [tex]\(g(x) = x - 4\)[/tex]
2. First, find [tex]\(g(10)\)[/tex]:
To find [tex]\(g(10)\)[/tex], we substitute [tex]\(10\)[/tex] into the function [tex]\(g(x)\)[/tex]:
[tex]\[
g(10) = 10 - 4 = 6
\][/tex]
3. Next, find [tex]\(f(g(10))\)[/tex] which is [tex]\(f(6)\)[/tex]:
Now, we substitute [tex]\(6\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[
f(6) = 6^2 + 1
\][/tex]
Calculate the square of [tex]\(6\)[/tex]:
[tex]\[
6^2 = 36
\][/tex]
Then add [tex]\(1\)[/tex]:
[tex]\[
36 + 1 = 37
\][/tex]
Therefore, the value of [tex]\((f \circ g)(10)\)[/tex] is [tex]\(37\)[/tex]. Thus, the correct answer is:
[tex]\[
\boxed{37}
\][/tex]
