Sure, let's work through the problem step by step to find [tex]\((f \circ g)(5)\)[/tex] where [tex]\(f(x) = x^2\)[/tex] and [tex]\(g(x) = x - 3\)[/tex].
1. First, understand the composition of functions: [tex]\((f \circ g)(x)\)[/tex] means applying [tex]\(g(x)\)[/tex] first and then applying [tex]\(f\)[/tex] to the result of [tex]\(g(x)\)[/tex].
2. Calculate [tex]\(g(5)\)[/tex]:
- Substitute 5 into [tex]\(g(x)\)[/tex]:
[tex]\[
g(5) = 5 - 3
\][/tex]
- Simplifying this gives:
[tex]\[
g(5) = 2
\][/tex]
3. Next, apply [tex]\(f\)[/tex] to the result of [tex]\(g(5)\)[/tex]:
- Substitute [tex]\(g(5)\)[/tex] which we found to be 2 into [tex]\(f(x)\)[/tex]:
[tex]\[
f(g(5)) = f(2)
\][/tex]
- Since [tex]\(f(x) = x^2\)[/tex], we compute [tex]\(f(2)\)[/tex]:
[tex]\[
f(2) = 2^2
\][/tex]
- Simplifying this gives:
[tex]\[
f(2) = 4
\][/tex]
Therefore, [tex]\((f \circ g)(5) = 4\)[/tex].
To summarize, we found that:
- [tex]\(g(5) = 2\)[/tex]
- [tex]\(f(g(5)) = 4\)[/tex]
Thus, [tex]\((f \circ g)(5) = 4\)[/tex].
