In this problem, we want to find the expression that represents [tex]\((t \circ s)(x)\)[/tex], where the function [tex]\(s(x) = x - 7\)[/tex] and the function [tex]\(t(x) = 4x^2 - x + 3\)[/tex].
The composition [tex]\((t \circ s)(x)\)[/tex] means we first apply the function [tex]\(s(x)\)[/tex] and then apply the function [tex]\(t(x)\)[/tex] on the result of [tex]\(s(x)\)[/tex].
1. Apply [tex]\(s(x)\)[/tex]:
[tex]\[
s(x) = x - 7
\][/tex]
2. Apply [tex]\(t\)[/tex] on [tex]\(s(x)\)[/tex]:
[tex]\[
t(s(x)) = t(x - 7)
\][/tex]
3. Evaluate [tex]\(t(x - 7)\)[/tex] by substituting [tex]\(x - 7\)[/tex] into the function [tex]\(t(x)\)[/tex]:
Since [tex]\(t(x) = 4x^2 - x + 3\)[/tex], we substitute [tex]\(x - 7\)[/tex] into [tex]\(t(x)\)[/tex]:
[tex]\[
t(x - 7) = 4(x - 7)^2 - (x - 7) + 3
\][/tex]
This is the expanded form. Let's simplify it step-by-step:
4. Square the binomial [tex]\((x - 7)\)[/tex]:
[tex]\[
(x - 7)^2 = x^2 - 14x + 49
\][/tex]
5. Substitute this back into the expression:
[tex]\[
t(x - 7) = 4(x^2 - 14x + 49) - (x - 7) + 3
\][/tex]
6. Distribute the 4:
[tex]\[
4(x^2 - 14x + 49) = 4x^2 - 56x + 196
\][/tex]
7. Combine all terms:
[tex]\[
t(x - 7) = 4x^2 - 56x + 196 - (x - 7) + 3
\][/tex]
Simplify the terms inside the expression:
[tex]\[
t(x - 7) = 4x^2 - 56x + 196 - x + 7 + 3
\][/tex]
Combine like terms:
[tex]\[
t(x - 7) = 4x^2 - 57x + 206
\][/tex]
None of the choices simplified to this. However, there seems to be a typo in the choices given; correct one among presented option is:
[tex]\[
t(x - 7) = 4(x - 7)^2 - (x - 7) + 3
\][/tex]
Thus, the correct simplified form given is:
[tex]\[
t(x - 7) = \boxed{4(x-7)^2-(x-7)+3}
\][/tex]
Notice that (x - 7)^2 can simplify further hence the final form is:
This matches one of the provided options precisely.
[tex]\[
\boxed{4(x - 7)^2 - (x - 7) + 3}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{4(x - 7)^2 - (x - 7) + 3}
\][/tex]
