To determine [tex]\((g \circ f)(x)\)[/tex], which means the composition of the functions [tex]\(g(x)\)[/tex] and [tex]\(f(x)\)[/tex], we need to apply [tex]\(f(x)\)[/tex] first and then apply [tex]\(g(x)\)[/tex] to the result of [tex]\(f(x)\)[/tex]. The given functions are:
[tex]\[f(x) = x + 4\][/tex]
[tex]\[g(x) = x^2 - 1\][/tex]
Here are the steps to find [tex]\((g \circ f)(x)\)[/tex]:
1. Apply [tex]\(f(x)\)[/tex]:
[tex]\[f(x) = x + 4\][/tex]
2. Substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[g(f(x)) = g(x + 4)\][/tex]
3. Evaluate [tex]\(g(x + 4)\)[/tex]:
[tex]\[g(x + 4) = (x + 4)^2 - 1\][/tex]
4. Expand [tex]\((x + 4)^2\)[/tex]:
[tex]\[
(x + 4)^2 = x^2 + 8x + 16
\][/tex]
5. Subtract 1 from the expanded result:
[tex]\[
(x + 4)^2 - 1 = x^2 + 8x + 16 - 1 = x^2 + 8x + 15
\][/tex]
Thus, the composition [tex]\((g \circ f)(x)\)[/tex] is:
[tex]\[
(g \circ f)(x) = x^2 + 8x + 15
\][/tex]
Therefore, the correct answer is:
[tex]\((g \circ f)(x) = x^2 + 8x + 15\)[/tex]
