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

First, find the value of the inner function: t ( − 7 ) = 3 × ( − 7 ) = − 21 .
Then, substitute this value into the outer function: s ( − 21 ) = 2 − ( − 21 ) 2 .
Simplify the expression: 2 − 441 = − 439 .
The final result 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 )) .

Evaluating 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

Evaluating 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

Simplifying the Expression Next, we simplify the expression: s ( − 21 ) = 2 − ( − 21 ) 2 = 2 − 441 = − 439

Final Answer Therefore, the value of ( s ∘ t ) ( − 7 ) is − 439 .


Examples
Composite functions are used in many real-world applications. For example, in manufacturing, a company might have a function that calculates the cost of producing x items, and another function that calculates the revenue from selling y items. By composing these functions, the company can determine the profit (revenue minus cost) as a function of the number of items produced. This helps in optimizing production levels to maximize profit. In computer graphics, composite functions are used to apply multiple transformations to an object, such as scaling, rotation, and translation, in a specific order.

Answered by GinnyAnswer | 2025-07-08