To determine which expression is equivalent to [tex]\((s t)(6)\)[/tex], let's analyze it step-by-step:
1. [tex]\((s t)(6)\)[/tex] means that we need to first understand the composition of the functions [tex]\(s\)[/tex] and [tex]\(t\)[/tex].
2. The expression indicates that we first apply the function [tex]\(t\)[/tex] to the input [tex]\(6\)[/tex], and then apply the function [tex]\(s\)[/tex] to the result of [tex]\(t(6)\)[/tex].
3. Mathematically, this can be represented as [tex]\(s(t(6))\)[/tex].
Let's match this understanding to the given options:
1. [tex]\(s(t(6))\)[/tex]: This precisely matches our interpretation. First [tex]\(t\)[/tex] is applied to [tex]\(6\)[/tex], and the result is then used as the input to [tex]\(s\)[/tex].
2. [tex]\(s(x) \cdot t(6)\)[/tex]: This represents multiplication of the function [tex]\(s(x)\)[/tex] with the result of [tex]\(t(6)\)[/tex]. This does not match our need to apply [tex]\(s\)[/tex] to [tex]\(t(6)\)[/tex].
3. [tex]\(s(6) \cdot t(6)\)[/tex]: This implies that both functions [tex]\(s\)[/tex] and [tex]\(t\)[/tex] are applied independently to [tex]\(6\)[/tex] and their results are multiplied. This is not the same as first applying [tex]\(t\)[/tex] to [tex]\(6\)[/tex] and then [tex]\(s\)[/tex] to the result.
4. [tex]\(6 \cdot s(x) \cdot t(x)\)[/tex]: This represents the multiplication of [tex]\(6\)[/tex] with both functions [tex]\(s(x)\)[/tex] and [tex]\(t(x)\)[/tex]. This does not correspond to applying [tex]\(s\)[/tex] to the result of [tex]\(t(6)\)[/tex].
Therefore, the expression equivalent to [tex]\((s t)(6)\)[/tex] is [tex]\(s(t(6))\)[/tex].
Thus, the correct option is:
1. [tex]\(s(t(6))\)[/tex]
