To determine the value of [tex]\((s \circ i)(-7)\)[/tex], we need to understand the composition of functions, specifically how to compose [tex]\(s(x)\)[/tex] and [tex]\(i(x)\)[/tex]. Here, [tex]\(i(x)\)[/tex] is the identity function, which means:
[tex]\[ i(x) = x \][/tex]
The composition of [tex]\(s\)[/tex] and [tex]\(i\)[/tex] is denoted as [tex]\((s \circ i)(x)\)[/tex], which means [tex]\(s(i(x))\)[/tex]. Since [tex]\(i(x) = x\)[/tex], this simplifies to:
[tex]\[ (s \circ i)(x) = s(i(x)) = s(x) \][/tex]
Therefore, [tex]\((s \circ i)(x)\)[/tex] is equivalent to simply [tex]\(s(x)\)[/tex].
Given:
[tex]\[ s(x) = 2 - x^2 \][/tex]
we substitute [tex]\(x = -7\)[/tex] into the function [tex]\(s(x)\)[/tex]:
[tex]\[ s(-7) = 2 - (-7)^2 \][/tex]
First, calculate [tex]\((-7)^2\)[/tex]:
[tex]\[ (-7)^2 = 49 \][/tex]
Then, substitute this value back into the function:
[tex]\[ s(-7) = 2 - 49 \][/tex]
Perform the subtraction:
[tex]\[ s(-7) = 2 - 49 = -47 \][/tex]
Thus, the value of [tex]\((s \circ i)(-7)\)[/tex] is [tex]\(-47\)[/tex].
The correct answer is not listed among the given options, but based on the calculations, [tex]\((s \circ i)(-7)\)[/tex] is equivalent to [tex]\(-47\)[/tex].
