To determine which expression is equivalent to [tex]\((t \circ s)(x)\)[/tex], let's recall that the composition of functions [tex]\((t \circ s)(x)\)[/tex] means [tex]\(t(s(x))\)[/tex]. This process involves substituting [tex]\(s(x)\)[/tex] into the function [tex]\(t(x)\)[/tex].
Given the functions:
[tex]\[ s(x) = x - 7 \][/tex]
[tex]\[ t(x) = 4x^2 - x + 3 \][/tex]
1. Substitute [tex]\(s(x)\)[/tex] into [tex]\(t(x)\)[/tex]:
[tex]\[ t(s(x)) = t(x - 7) \][/tex]
2. We need to substitute [tex]\(x - 7\)[/tex] for every [tex]\(x\)[/tex] in [tex]\(t(x)\)[/tex]:
[tex]\[ t(x-7) = 4(x-7)^2 - (x-7) + 3 \][/tex]
3. Simplify [tex]\(t(x-7)\)[/tex]:
[tex]\[
t(x-7) = 4(x-7)^2 - (x-7) + 3
\][/tex]
Expanding [tex]\( (x-7)^2 \)[/tex]:
[tex]\[
(x-7)^2 = x^2 - 14x + 49
\][/tex]
Now substitute this back into the expression:
[tex]\[
4(x-7)^2 = 4(x^2 - 14x + 49) = 4x^2 - 56x + 196
\][/tex]
Thus:
[tex]\[
t(x-7) = 4(x^2 - 14x + 49) - (x - 7) + 3
\][/tex]
[tex]\[
= 4x^2 - 56x + 196 - x + 7 + 3
\][/tex]
Combine like terms:
[tex]\[
= 4x^2 - 57x + 206
\][/tex]
That final expression matches with:
[tex]\[ -x + 4(x - 7)^2 + 10 \][/tex]
which simplifies back to:
[tex]\[ 4(x-7)^2 - (x-7) + 3 \][/tex]
Thus, the expression that is equivalent to [tex]\((t \circ s)(x)\)[/tex] is:
[tex]\[ \boxed{4(x-7)^2 - (x-7) + 3} \][/tex]
