To determine the expression equivalent to [tex]\((g \circ h)(5)\)[/tex], we need to find what [tex]\(h(5)\)[/tex] is first, and then apply [tex]\(g\)[/tex] to the result of [tex]\(h(5)\)[/tex].
Given functions:
[tex]\(h(x) = x - 7\)[/tex]
[tex]\(g(x) = x^2\)[/tex]
1. Calculate [tex]\(h(5)\)[/tex]:
[tex]\[ h(5) = 5 - 7 = -2 \][/tex]
2. Next, apply [tex]\(g\)[/tex] to [tex]\(h(5)\)[/tex] which means we need [tex]\(g(-2)\)[/tex]:
[tex]\[ g(-2) = (-2)^2 = 4 \][/tex]
Thus, [tex]\( (g \circ h)(5) = g(h(5)) = g(-2) = 4 \)[/tex].
Now, let's match this to the given expressions:
- Option 1: [tex]\((5-7)^2\)[/tex]
[tex]\[ (5-7)^2 = (-2)^2 = 4 \][/tex]
- Option 2: [tex]\((5)^2 - 7\)[/tex]
[tex]\[ 5^2 - 7 = 25 - 7 = 18 \][/tex]
- Option 3: [tex]\((5)^2(5-7)\)[/tex]
[tex]\[ 5^2(5-7) = 25 \times (-2) = -50 \][/tex]
- Option 4: [tex]\((5-7) x^2\)[/tex]
This expression doesn't conform to typical notation for functions and is not a relevant mathematical transformation.
Therefore, the equivalent expression to [tex]\((g \circ h)(5)\)[/tex] is:
[tex]\[ (5-7)^2 \][/tex]
This matches the calculation we did for [tex]\((g \circ h)(5)\)[/tex]. The correct answer is:
[tex]\[ (5-7)^2 \][/tex]
