If $s(x)=x-7$ and $t(x)=4 x^2-x+3$, which expression is equivalent to $(t \circ s)(x)$?
A. $4(x-7)^2-x-7+3$
B. $4(x-7)^2-(x-7)+3$
C. $(4 x^2-x+3)-7$
D. $(4 x^2-x+3)(x-7)$
Answer (1)
Substitute s ( x ) = x − 7 into t ( x ) = 4 x 2 − x + 3 .
Obtain the composite function t ( s ( x )) = 4 ( x − 7 ) 2 − ( x − 7 ) + 3 .
The equivalent expression for ( t c i rcs ) ( x ) 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 ) . This means we need to find t ( s ( x )) , which involves substituting s ( x ) into t ( x ) .
Substituting s(x) into t(x) To find ( t ∘ s ) ( x ) , we substitute s ( x ) = x − 7 into the function t ( x ) = 4 x 2 − x + 3 . This means we replace every instance of x in t ( x ) with ( x − 7 ) .
Finding the Composite Function So, we have t ( s ( x )) = t ( x − 7 ) = 4 ( x − 7 ) 2 − ( x − 7 ) + 3 . This is the expression we are looking for.
Identifying the Correct Option Now, we compare our expression 4 ( x − 7 ) 2 − ( x − 7 ) + 3 with the given options to find the matching one. The correct option is 4 ( x − 7 ) 2 − ( x − 7 ) + 3 .
Final Answer Therefore, the expression equivalent to ( t ∘ s ) ( x ) is 4 ( x − 7 ) 2 − ( x − 7 ) + 3 .
Examples
Function composition is a fundamental concept in mathematics with applications in various fields. For instance, in computer graphics, transformations like scaling, rotation, and translation can be represented as functions. Composing these functions allows us to combine multiple transformations into a single function, making it easier to apply complex transformations to objects. Similarly, in physics, the position of an object can be expressed as a function of time, and the velocity can be expressed as a function of position. Composing these functions allows us to determine the velocity of the object at a specific time.
