To determine which expression is equivalent to [tex]\((st)(6)\)[/tex], we need to understand the composition of functions notation. The notation [tex]\((st)(6)\)[/tex] indicates that two functions [tex]\(s\)[/tex] and [tex]\(t\)[/tex] are composed, and the result of this composition is then applied to the number 6.
Here are the steps to solve this:
1. Understanding Function Composition:
- When we write [tex]\((st)\)[/tex], it means [tex]\(s(t(x))\)[/tex], which is the function [tex]\(s\)[/tex] applied to the result of the function [tex]\(t\)[/tex] applied to [tex]\(x\)[/tex].
- Thus, [tex]\((st)(6)\)[/tex] means [tex]\(s(t(6))\)[/tex].
2. Analyzing the Options:
- [tex]\(s(t(6))\)[/tex]: This is exactly the same as [tex]\((st)(6)\)[/tex] since it applies [tex]\(t\)[/tex] to 6 first and then applies [tex]\(s\)[/tex] to the result.
- [tex]\(s(x) \times t(6)\)[/tex]: This suggests multiplying the function [tex]\(s\)[/tex] applied to a variable [tex]\(x\)[/tex] with the function [tex]\(t\)[/tex] applied to 6. This does not match the required composition.
- [tex]\(s(6) \times t(6)\)[/tex]: This indicates that [tex]\(s\)[/tex] is applied to 6 and [tex]\(t\)[/tex] is applied to 6, and then the results are multiplied together. This is not the same as first applying [tex]\(t\)[/tex] to 6 and then applying [tex]\(s\)[/tex] to the result.
- [tex]\(6 \times s(x) \times t(x)\)[/tex]: This expression involves multiplying 6 with the product of [tex]\(s\)[/tex] and [tex]\(t\)[/tex] applied to a general variable [tex]\(x\)[/tex], which is completely different from the composition of functions on 6.
3. Conclusion:
- The only option that accurately represents the composition [tex]\(s(t(6))\)[/tex] is the first option.
Thus, the expression equivalent to [tex]\((st)(6)\)[/tex] is:
[tex]\[
\boxed{s(t(6))}
\][/tex]
