Let's solve the problem step-by-step.
1. Understand the definition of functions [tex]\(s(x)\)[/tex] and [tex]\(t(x)\)[/tex]:
[tex]\[
s(x) = x - 7
\][/tex]
[tex]\[
t(x) = 4x^2 - x + 3
\][/tex]
2. Understand the composition [tex]\((t \cdot s)(x)\)[/tex]:
[tex]\((t \cdot s)(x)\)[/tex] means applying the function [tex]\( t \)[/tex] to the result of [tex]\( s(x) \)[/tex]:
[tex]\[
(t \cdot s)(x) = t(s(x))
\][/tex]
3. Substitute [tex]\(s(x)\)[/tex] into [tex]\(t(x)\)[/tex]:
First evaluate [tex]\(s(x)\)[/tex]:
[tex]\[
s(x) = x - 7
\][/tex]
Now, substitute [tex]\(s(x)\)[/tex] into [tex]\(t(x)\)[/tex]:
[tex]\[
t(s(x)) = t(x - 7)
\][/tex]
Replace [tex]\(x\)[/tex] with [tex]\(x - 7\)[/tex] in the function [tex]\(t(x)\)[/tex]:
[tex]\[
t(x - 7) = 4(x - 7)^2 - (x - 7) + 3
\][/tex]
4. Simplify the expression [tex]\((t \cdot s)(x)\)[/tex]:
[tex]\[
t(x - 7) = 4(x - 7)^2 - (x - 7) + 3
\][/tex]
Let's verify which option this agrees with:
[tex]\[
\text{Option 1: } 4(x-7)^2 - x - 7 + 3
\][/tex]
[tex]\[
\text{Option 2: } 4(x-7)^2 - (x-7) + 3
\][/tex]
[tex]\[
\text{Option 3: } (4x^2 - x + 3) - 7
\][/tex]
[tex]\[
\text{Option 4: } (4x^2 - x + 3)(x-7)
\][/tex]
5. Compare the simplified expression with the given options:
The simplified expression [tex]\(4(x - 7)^2 - (x - 7) + 3\)[/tex] directly matches with Option 2.
So, the correct equivalent expression is:
[tex]\[
4(x-7)^2 - (x-7) + 3
\][/tex]
