Answer :
To determine which expression is equivalent to [tex]\((s t)(6)\)[/tex], let's analyze the provided options.
1. Expression: [tex]\(s(t(6))\)[/tex]
Here, [tex]\(t(6)\)[/tex] is calculated first, and then [tex]\(s\)[/tex] is applied to this result. We can represent this process by considering the composition of functions where [tex]\(s\)[/tex] acts on the result of [tex]\(t(6)\)[/tex]. This matches the form [tex]\((s \circ t)(6)\)[/tex].
2. Expression: [tex]\(s(x) \times t(6)\)[/tex]
In this expression, [tex]\(t(6)\)[/tex] is calculated first. However, [tex]\(s(x)\)[/tex] is simply a multiplication by the value of [tex]\(s\)[/tex] at some general [tex]\(x\)[/tex], not specifically related to [tex]\(t(6)\)[/tex]. This does not represent function composition, but rather a product of two separate evaluations.
3. Expression: [tex]\(s(6) \times t(6)\)[/tex]
Here, both [tex]\(s\)[/tex] and [tex]\(t\)[/tex] are evaluated independently at [tex]\(6\)[/tex], and the results are multiplied. This is still not the composition of functions but rather independent evaluations multiplied together.
4. Expression: [tex]\(6 \times s(x) \times t(x)\)[/tex]
In this expression, the number [tex]\(6\)[/tex] is multiplied by the values of [tex]\(s(x)\)[/tex] and [tex]\(t(x)\)[/tex] for some general [tex]\(x\)[/tex]. This does not involve applying one function to the result of another and is far from representing function composition.
Given the explanations above, the correct expression that represents [tex]\((s t)(6)\)[/tex] in terms of function composition is:
[tex]\[ \boxed{s(t(6))} \][/tex]
1. Expression: [tex]\(s(t(6))\)[/tex]
Here, [tex]\(t(6)\)[/tex] is calculated first, and then [tex]\(s\)[/tex] is applied to this result. We can represent this process by considering the composition of functions where [tex]\(s\)[/tex] acts on the result of [tex]\(t(6)\)[/tex]. This matches the form [tex]\((s \circ t)(6)\)[/tex].
2. Expression: [tex]\(s(x) \times t(6)\)[/tex]
In this expression, [tex]\(t(6)\)[/tex] is calculated first. However, [tex]\(s(x)\)[/tex] is simply a multiplication by the value of [tex]\(s\)[/tex] at some general [tex]\(x\)[/tex], not specifically related to [tex]\(t(6)\)[/tex]. This does not represent function composition, but rather a product of two separate evaluations.
3. Expression: [tex]\(s(6) \times t(6)\)[/tex]
Here, both [tex]\(s\)[/tex] and [tex]\(t\)[/tex] are evaluated independently at [tex]\(6\)[/tex], and the results are multiplied. This is still not the composition of functions but rather independent evaluations multiplied together.
4. Expression: [tex]\(6 \times s(x) \times t(x)\)[/tex]
In this expression, the number [tex]\(6\)[/tex] is multiplied by the values of [tex]\(s(x)\)[/tex] and [tex]\(t(x)\)[/tex] for some general [tex]\(x\)[/tex]. This does not involve applying one function to the result of another and is far from representing function composition.
Given the explanations above, the correct expression that represents [tex]\((s t)(6)\)[/tex] in terms of function composition is:
[tex]\[ \boxed{s(t(6))} \][/tex]
