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If [tex][tex]$h(x)=x-7$[/tex][/tex] and [tex][tex]$g(x)=x^2$[/tex][/tex], which expression is equivalent to [tex][tex]$(g \circ h)(5)$[/tex][/tex]?

A. [tex][tex]$(5-7)^2$[/tex][/tex]
B. [tex][tex]$(5)^2-7$[/tex][/tex]
C. [tex][tex]$(5)^2(5-7)$[/tex][/tex]
D. [tex][tex]$(5-7) x^2$[/tex][/tex]

Sagot :

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]