Let's evaluate each of the given expressions step-by-step to see which one is equivalent to [tex]\((s \circ t)(6)\)[/tex].
First, understand that [tex]\((s \circ t)(x)\)[/tex] means [tex]\(s(t(x))\)[/tex], which represents applying the function [tex]\(t\)[/tex] on [tex]\(x\)[/tex] first, and then applying the function [tex]\(s\)[/tex] on the result of [tex]\(t(x)\)[/tex].
Given that [tex]\(s\)[/tex] and [tex]\(t\)[/tex] are identity functions for any input [tex]\(x\)[/tex]:
[tex]\[ s(x) = x \][/tex]
[tex]\[ t(x) = x \][/tex]
Let's evaluate the expressions one by one:
1. [tex]\(s(t(6))\)[/tex]:
[tex]\[
s(t(6)) = s(6) = 6
\][/tex]
So, this expression equals 6.
2. [tex]\(s(x) \cdot t(6)\)[/tex]:
[tex]\[
s(x) \cdot t(6) = s(6) \cdot 6 = 6 \cdot 6 = 36
\][/tex]
So, this expression equals 36.
3. [tex]\(s(6) \cdot t(6)\)[/tex]:
[tex]\[
s(6) \cdot t(6) = 6 \cdot 6 = 36
\][/tex]
So, this expression equals 36.
4. [tex]\(6 \cdot s(x) \cdot t(x)\)[/tex]:
[tex]\[
6 \cdot s(6) \cdot t(6) = 6 \cdot 6 \cdot 6 = 216
\][/tex]
So, this expression equals 216.
After evaluating all the expressions, it's clear that the expression equivalent to [tex]\((s \circ t)(6)\)[/tex] is:
[tex]\[ s(t(6)) = 6 \][/tex]
Therefore, the correct expression is:
[tex]\[ s(t(6)) \][/tex]
