To determine [tex]\((g \circ f)(x)\)[/tex] when [tex]\(f(x) = \sqrt{x} - 3\)[/tex] and [tex]\(g(x) = x - 7\)[/tex], we need to find the composition of the functions, which is essentially [tex]\(g(f(x))\)[/tex].
Let's go through this step-by-step:
1. Evaluate [tex]\(f(x)\)[/tex]:
[tex]\(f(x) = \sqrt{x} - 3\)[/tex]
2. Substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
First, replace [tex]\(x\)[/tex] in [tex]\(g(x)\)[/tex] with [tex]\(f(x)\)[/tex]:
[tex]\(g(f(x))\)[/tex]
3. Substitute and simplify:
Since [tex]\(f(x) = \sqrt{x} - 3\)[/tex], we substitute this into [tex]\(g\)[/tex]:
[tex]\(g(\sqrt{x} - 3)\)[/tex]
Now, replace the argument in [tex]\(g\)[/tex]:
[tex]\(g(\sqrt{x} - 3) = (\sqrt{x} - 3) - 7\)[/tex]
Simplify the expression inside [tex]\(g\)[/tex]:
[tex]\[
(\sqrt{x} - 3) - 7 = \sqrt{x} - 10
\][/tex]
Therefore, the composition [tex]\((g \circ f)(x)\)[/tex] is:
[tex]\[
(g \circ f)(x) = \sqrt{x} - 10
\][/tex]
Hence, the correct answer is:
[tex]\[
(g \circ f)(x) = \sqrt{x} - 10
\][/tex]
