If [tex]s(x)=x-7[/tex] and [tex]t(x)=4 x^2-x+3[/tex], which expression is equivalent to [tex](t \circ s)(x)[/tex]?
A. [tex]4(x-7)^2-x-7+3[/tex]
B. [tex]4(x-7)^2-(x-7)+3[/tex]
C. [tex](4 x^2-x+3)-7[/tex]
D. [tex](4 x^2-x+3)(x-7)[/tex]
Answer (1)
Substitute s ( x ) = x − 7 into t ( x ) = 4 x 2 − x + 3 .
Obtain t ( s ( x )) = 4 ( x − 7 ) 2 − ( x − 7 ) + 3 .
The equivalent expression is 4 ( x − 7 ) 2 − ( x − 7 ) + 3 .
Explanation
Understanding the Problem We are given two functions, s ( x ) = x − 7 and t ( x ) = 4 x 2 − x + 3 . We want to find the expression that is equivalent to the composition ( t ∘ s ) ( x ) , which means t ( s ( x )) .
Composition of Functions To find ( t ∘ s ) ( x ) , we need to substitute the function s ( x ) into the function t ( x ) wherever we see x . In other words, we need to evaluate t ( x − 7 ) .
Substitution We have t ( x ) = 4 x 2 − x + 3 . So, we replace x with ( x − 7 ) in the expression for t ( x ) : t ( x − 7 ) = 4 ( x − 7 ) 2 − ( x − 7 ) + 3
Finding the Equivalent Expression Now, we compare this expression with the given options to find the equivalent expression. The expression we found is 4 ( x − 7 ) 2 − ( x − 7 ) + 3 , which matches the second option.
Final Answer Therefore, the expression equivalent to ( t ∘ s ) ( x ) is 4 ( x − 7 ) 2 − ( x − 7 ) + 3 .
Examples
Understanding function composition is crucial in many areas, such as physics and computer science. For example, in physics, if s ( t ) represents the position of an object at time t , and t ( v ) represents the time it takes to reach a certain velocity v , then ( s ∘ t ) ( v ) would represent the position of the object when it reaches velocity v . In computer graphics, transformations like scaling, rotation, and translation can be represented as functions, and applying multiple transformations is equivalent to composing these functions. This allows for complex animations and visual effects to be created by combining simpler transformations.
