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)
The value of ( s ∘ t ) ( − 7 ) is − 439 , which is option A. This is found by first calculating the value of t ( − 7 ) and then substituting that result into s ( x ) . The final computation gives us the answer of − 439 .
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First, find t ( − 7 ) by substituting − 7 into t ( x ) : t ( − 7 ) = 3 × ( − 7 ) = − 21 .
Then, substitute t ( − 7 ) into s ( x ) : s ( − 21 ) = 2 − ( − 21 ) 2 .
Calculate ( − 21 ) 2 = 441 .
Finally, compute s ( − 21 ) = 2 − 441 = − 439 , so the answer is − 439 .
Explanation
Understanding the Problem We are given two functions, s ( x ) = 2 − x 2 and t ( x ) = 3 x . We want to find the value of the composite function ( s ∘ t ) ( − 7 ) . This 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 function s ( x ) to find s ( t ( − 7 )) :
s ( t ( − 7 )) = s ( − 21 ) = 2 − ( − 21 ) 2 = 2 − 441 = − 439
Final Answer Therefore, ( s ∘ t ) ( − 7 ) = − 439 .
Examples
Composite functions are useful in many real-world scenarios. For example, consider a store that offers a discount of 10% on all items and then applies a sales tax of 5%. If x is the original price, the discounted price is d ( x ) = 0.9 x , and the price after tax is t ( x ) = 1.05 x . The final price after both discount and tax is applied can be represented by the composite function ( t ∘ d ) ( x ) = t ( d ( x )) = 1.05 ( 0.9 x ) = 0.945 x . This shows how composite functions can model sequential operations.
