To find which expression is equivalent to [tex]\((g \circ h)(5)\)[/tex], let's break down the steps:
1. Understanding the function composition [tex]\((g \circ h)(5)\)[/tex]:
- Here, [tex]\(h(x) = x - 7\)[/tex]
- And [tex]\(g(x) = x^2\)[/tex]
2. First, we need to evaluate [tex]\(h(5)\)[/tex]:
- Plug [tex]\(x = 5\)[/tex] into [tex]\(h(x)\)[/tex]
- [tex]\(h(5) = 5 - 7 = -2\)[/tex]
3. Next, we evaluate [tex]\(g(h(5))\)[/tex]:
- We found [tex]\(h(5) = -2\)[/tex]
- Now, [tex]\(g(-2)\)[/tex]: Substitute [tex]\(-2\)[/tex] into [tex]\(g(x)\)[/tex]
- [tex]\(g(-2) = (-2)^2 = 4\)[/tex]
4. Determine the equivalent expression for the composition [tex]\((g \circ h)(5)\)[/tex]:
- We need to see how the computed value [tex]\((g(h(5))\)[/tex] relates to the given expressions.
Let's check the given options one by one:
1. [tex]\((5-7)^2\)[/tex]:
- Calculate [tex]\((5 - 7)^2\)[/tex]
- [tex]\(5 - 7 = -2\)[/tex]
- [tex]\((-2)^2 = 4\)[/tex]
- This matches our computed value for [tex]\(g(h(5))\)[/tex]
2. [tex]\((5)^2 - 7\)[/tex]:
- Calculate [tex]\((5)^2 - 7\)[/tex]
- [tex]\(5^2 = 25\)[/tex]
- [tex]\(25 - 7 = 18\)[/tex]
- This does not match our computed value
3. [tex]\((5 - 7) x^2\)[/tex]:
- Calculate [tex]\((5 - 7) x^2\)[/tex]
- [tex]\(5 - 7 = -2\)[/tex]
- This means we are looking for [tex]\(-2 x^2\)[/tex]
- This is not the same as [tex]\((5 - 7)^2\)[/tex]
4. [tex]\((5)^2 (5 - 7)\)[/tex]:
- Calculate [tex]\((5)^2 (5 - 7)\)[/tex]
- [tex]\(5^2 = 25\)[/tex]
- [tex]\(5 - 7 = -2\)[/tex]
- [tex]\(25 \times (-2) = -50\)[/tex]
- This does not match our computed value
From the above checks, the expression equivalent to [tex]\((g \circ h)(5)\)[/tex] is:
[tex]\((5-7)^2\)[/tex]
So, the answer is [tex]\((5-7)^2\)[/tex].
