To solve this problem, we need to find the value of [tex]\((s \circ t)(-7)\)[/tex]. This means we need to first apply the function [tex]\( t(x) \)[/tex] to the input [tex]\(-7\)[/tex] and then apply the function [tex]\( s(x) \)[/tex] to the result. Here is the step-by-step procedure:
1. Apply the function [tex]\( t(x) \)[/tex] to [tex]\(-7\)[/tex]:
Start with [tex]\( t(x) = 3x \)[/tex].
Plug in [tex]\( x = -7 \)[/tex]:
[tex]\[
t(-7) = 3 \times (-7) = -21
\][/tex]
2. Now, apply the function [tex]\( s(x) \)[/tex] to the result of [tex]\( t(-7) \)[/tex]:
The result from the first step is [tex]\(-21\)[/tex], so we need to find [tex]\( s(-21) \)[/tex].
Use the function [tex]\( s(x) = 2 - x^2 \)[/tex].
Plug in [tex]\( x = -21 \)[/tex]:
[tex]\[
s(-21) = 2 - (-21)^2
\][/tex]
3. Calculate the value:
First, calculate [tex]\( (-21)^2 \)[/tex]:
[tex]\[
(-21)^2 = 441
\][/tex]
Next, substitute this back into the function [tex]\( s(x) \)[/tex]:
[tex]\[
s(-21) = 2 - 441 = -439
\][/tex]
Therefore, the value of [tex]\((s \circ t)(-7)\)[/tex] is [tex]\(-439\)[/tex].
From the given options, the correct answer is [tex]\(-439\)[/tex].
