If [tex]c(x)=4 x-2[/tex] and [tex]d(x)=x^2+5 x[/tex], what is [tex](c \circ d)(x)[/tex]?
A. [tex]4 x^3+18 x^2-10 x[/tex]
B. [tex]x^2+9 x-2[/tex]
C. [tex]16 x^2+4 x-6[/tex]
D. [tex]4 x^2+20 x-2[/tex]
Answer (2)
Substitute d ( x ) into c ( x ) : ( c ∘ d ) ( x ) = c ( d ( x )) = c ( x 2 + 5 x ) .
Replace x in c ( x ) with ( x 2 + 5 x ) : c ( x 2 + 5 x ) = 4 ( x 2 + 5 x ) − 2 .
Expand the expression: 4 ( x 2 + 5 x ) − 2 = 4 x 2 + 20 x − 2 .
The composite function is 4 x 2 + 20 x − 2 .
Explanation
Understanding the Problem We are given two functions, c ( x ) = 4 x − 2 and d ( x ) = x 2 + 5 x . Our goal is to find the composite function ( c ∘ d ) ( x ) , which means we need to evaluate c ( d ( x )) . In other words, we will substitute d ( x ) into c ( x ) wherever we see x .
Substituting d(x) into c(x) To find ( c ∘ d ) ( x ) , we substitute d ( x ) into c ( x ) . So we have:
( c ∘ d ) ( x ) = c ( d ( x )) = c ( x 2 + 5 x )
Now, we replace x in the expression for c ( x ) with ( x 2 + 5 x ) :
c ( x 2 + 5 x ) = 4 ( x 2 + 5 x ) − 2
Expanding the Expression Next, we expand the expression:
4 ( x 2 + 5 x ) − 2 = 4 x 2 + 20 x − 2
So, ( c ∘ d ) ( x ) = 4 x 2 + 20 x − 2 .
Final Answer Therefore, the composite function ( c ∘ d ) ( x ) is 4 x 2 + 20 x − 2 .
Examples
Composite functions are useful in many real-world scenarios. For example, suppose a store is offering a discount of 20% on all items, and you also have a coupon for 10 o ff . I f x i s t h eor i g ina lp r i ceo f ani t e m , t h e p r i ce a f t er t h e 20 d(x) = 0.8x$, and the price after the 10 co u p o ni s c(x) = x - 10 . T h eco m p os i t e f u n c t i o n (c \circ d)(x) = c(d(x)) = 0.8x - 10$ represents the final price if the discount is applied first, followed by the coupon. Understanding composite functions helps you determine the best order to apply discounts and coupons to minimize the final price.
The composite function (c d)(x) = 4x^2 + 20x - 2 can be found by substituting d ( x ) into c ( x ) . After substituting and expanding, we see that the correct answer choice is D: 4 x 2 + 20 x − 2 .
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