If [tex]s(x)=2-x^2[/tex] and [tex]t(x)=3 x[/tex], which value is equivalent to [tex](s \circ t)(-7)[/tex]?
A. -439
B. -141
C. 153
D. 443
Answer (2)
First, find t ( − 7 ) by substituting − 7 into t ( x ) : t ( − 7 ) = 3 × ( − 7 ) = − 21 .
Then, find s ( t ( − 7 )) by substituting − 21 into s ( x ) : s ( − 21 ) = 2 − ( − 21 ) 2 = 2 − 441 = − 439 .
Thus, ( s ∘ t ) ( − 7 ) = − 439 .
The final answer is − 439 .
Explanation
Understanding the Problem We are given two functions, s ( x ) = 2 − x 2 and t ( x ) = 3 x . We need to find the value of the composite function ( s ∘ t ) ( − 7 ) , which means we need to evaluate s ( t ( − 7 )) .
Calculating t(-7) First, we need to find the value of t ( − 7 ) . We substitute x = − 7 into the expression for t ( x ) : t ( − 7 ) = 3 × ( − 7 ) = − 21
Calculating s(t(-7)) Now, we substitute the result, t ( − 7 ) = − 21 , into the expression for s ( x ) :
s ( t ( − 7 )) = s ( − 21 ) = 2 − ( − 21 ) 2 = 2 − 441 = − 439
Final Answer Therefore, ( s ∘ t ) ( − 7 ) = − 439 .
Examples
Composite functions are used in many real-world applications. For example, in manufacturing, one function might describe the cost of producing x items, and another function might describe the number of items produced as a function of time. Combining these functions allows you to determine the cost of production as a function of time. Similarly, in physics, you might have one function that describes the position of an object as a function of time, and another function that describes the velocity of the object as a function of position. Combining these functions allows you to determine the velocity of the object as a function of time.
The composite function ( s ∘ t ) ( − 7 ) evaluates to − 439 after calculating t ( − 7 ) and substituting that result into s ( x ) . The steps involve evaluating t ( − 7 ) to be − 21 and then finding s ( − 21 ) . Hence, the final answer is − 439 .
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