To solve the given problem, we need to find the value of the composite function [tex]\((g \circ h)(5)\)[/tex], given the functions [tex]\(h(x)\)[/tex] and [tex]\(g(x)\)[/tex].
1. We start with the inner function [tex]\(h(x)\)[/tex] which is [tex]\(h(x) = x - 7\)[/tex].
- Evaluate [tex]\(h(5)\)[/tex]:
[tex]\[
h(5) = 5 - 7 = -2
\][/tex]
2. Next, we use the result from [tex]\(h(5)\)[/tex] to find [tex]\(g(h(5))\)[/tex]. The outer function [tex]\(g(x)\)[/tex] is [tex]\(g(x) = x^2\)[/tex].
- Substitute the value obtained from [tex]\(h(5)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[
g(-2) = (-2)^2 = 4
\][/tex]
Thus, [tex]\((g \circ h)(5) = g(h(5)) = g(-2) = 4\)[/tex].
Now, we check which of the given expressions is equivalent to this computation:
1. [tex]\((5-7)^2\)[/tex]:
[tex]\[
(5 - 7)^2 = (-2)^2 = 4
\][/tex]
2. [tex]\((5)^2 - 7\)[/tex]:
[tex]\[
5^2 - 7 = 25 - 7 = 18
\][/tex]
3. [tex]\((5)^2(5-7)\)[/tex]:
[tex]\[
5^2 (5 - 7) = 25 \times (-2) = -50
\][/tex]
4. [tex]\((5-7) x^2\)[/tex]:
[tex]\[
(5 - 7) x^2 = -2 x^2 \text{ (This expression has an x variable and does not match our requirement.)}
\][/tex]
Since [tex]\( (g \circ h)(5) = 4 \)[/tex], the correct equivalent expression is:
[tex]\[
(5 - 7)^2
\][/tex]
Thus, the correct expression is [tex]\((5-7)^2\)[/tex].
