To determine the value of [tex]\((s \circ t)(-7)\)[/tex], we first need to understand the composition of functions. The composition [tex]\( (s \circ t)(x) \)[/tex] means that we apply the function [tex]\( t(x) \)[/tex] first and then plug the result into the function [tex]\( s(x) \)[/tex].
Given the functions:
[tex]\[ s(x) = 2 - x^2 \][/tex]
[tex]\[ t(x) = 3x \][/tex]
We need to perform the following steps:
1. Evaluate [tex]\( t(-7) \)[/tex].
2. Substitute the result of [tex]\( t(-7) \)[/tex] into [tex]\( s(x) \)[/tex].
Step 1: Evaluate [tex]\( t(-7) \)[/tex]
[tex]\[ t(-7) = 3 \cdot (-7) = -21 \][/tex]
Step 2: Substitute [tex]\( t(-7) \)[/tex] into [tex]\( s(x) \)[/tex]
Now we need to find [tex]\( s(t(-7)) \)[/tex], which is [tex]\( s(-21) \)[/tex].
[tex]\[ s(-21) = 2 - (-21)^2 \][/tex]
Next, calculate [tex]\( (-21)^2 \)[/tex]:
[tex]\[ (-21)^2 = 441 \][/tex]
Then, substitute this value back into [tex]\( s(x) \)[/tex]:
[tex]\[ s(-21) = 2 - 441 = -439 \][/tex]
Therefore, the value equivalent to [tex]\((s \circ t)(-7)\)[/tex] is [tex]\( -439 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{-439} \][/tex]
