To solve for the composite function [tex]\((f \circ g)(x)\)[/tex], we need to understand what it means to compose these two functions. This involves substituting the output of [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex].
Given:
[tex]\[ f(x) = x^2 - 2 \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]
The composite function [tex]\((f \circ g)(x)\)[/tex] is defined as:
[tex]\[ (f \circ g)(x) = f(g(x)) \][/tex]
Let's find [tex]\(f(g(x))\)[/tex]:
1. First, determine [tex]\(g(x)\)[/tex]:
[tex]\[ g(x) = x - 3 \][/tex]
2. Next, substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(x)) = f(x - 3) \][/tex]
3. Substitute [tex]\(x - 3\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(x - 3) = (x - 3)^2 - 2 \][/tex]
4. Expand [tex]\((x - 3)^2\)[/tex]:
[tex]\[ (x - 3)^2 = x^2 - 6x + 9 \][/tex]
5. Substitute the expanded form back into the expression:
[tex]\[ f(x - 3) = x^2 - 6x + 9 - 2 \][/tex]
6. Simplify the expression by combining like terms:
[tex]\[ f(x - 3) = x^2 - 6x + 7 \][/tex]
So, the composite function [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[ (f \circ g)(x) = x^2 - 6x + 7 \][/tex]
Therefore, the correct answer is:
[tex]\[ (f \circ g)(x) = x^2 - 6x + 7 \][/tex]
Hence, the correct choice is:
[tex]\[ (f \circ g)(x) = x^2 - 6x + 7 \][/tex]
