To determine which expression is equivalent to [tex]\((t \circ s)(x)\)[/tex], we begin by interpreting the composite function [tex]\((t \circ s)(x)\)[/tex]. This notation means that we need to substitute [tex]\(s(x)\)[/tex] into [tex]\(t(x)\)[/tex].
Given:
[tex]\[ s(x) = x - 7 \][/tex]
[tex]\[ t(x) = 4x^2 - x + 3 \][/tex]
We want to find [tex]\( t(s(x)) \)[/tex], which requires substituting [tex]\(s(x) = x - 7\)[/tex] into the function [tex]\(t(x)\)[/tex]. So let's break this down step-by-step:
1. Start by substituting [tex]\( s(x) \)[/tex] into [tex]\( t(x) \)[/tex]:
[tex]\[ t(s(x)) = t(x - 7) \][/tex]
2. Substitute [tex]\( x - 7 \)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( t(x) \)[/tex]:
[tex]\[ t(x - 7) = 4(x - 7)^2 - (x - 7) + 3 \][/tex]
3. Simplify the expression we get from the substitution:
[tex]\[ t(x - 7) = 4(x - 7)^2 - (x - 7) + 3 \][/tex]
4. We expand the squared term [tex]\( (x - 7)^2 \)[/tex]:
[tex]\[ (x - 7)^2 = x^2 - 14x + 49 \][/tex]
5. Substitute this expansion back into the equation:
[tex]\[ t(x - 7) = 4(x^2 - 14x + 49) - (x - 7) + 3 \][/tex]
6. Distribute the 4 in the expansion:
[tex]\[ t(x - 7) = 4x^2 - 56x + 196 - (x - 7) + 3 \][/tex]
7. Combine the terms inside the expression:
[tex]\[ t(x - 7) = 4x^2 - 56x + 196 - x + 7 + 3 \][/tex]
8. Finally, simplify the constants:
[tex]\[ t(x - 7) = 4x^2 - 57x + 206 \][/tex]
From this simplification, we see that the expression equivalent to [tex]\( (t \circ s)(x) \)[/tex] is indeed:
[tex]\[ 4(x-7)^2 - (x-7) + 3 \][/tex]
So, the correct answer is:
[tex]\[ 4(x-7)^2 - (x-7) + 3 \][/tex]
This matches the second option provided:
[tex]\[ 4(x-7)^2-(x-7)+3 \][/tex]
