If $c(x)=4x-2$ and $d(x)=x^2+5x$, what is $(c \circ d)(x)$?
A. $4x^3+18x^2-10x$
B. $x^2+9x-2$
C. $16x^2+4x-6$
D. $4x^2+20x-2$
Answer (2)
Substitute d ( x ) into c ( x ) : c ( d ( x )) = 4 ( x 2 + 5 x ) − 2 .
Expand the expression: 4 x 2 + 20 x − 2 .
Simplify to find the composite function: ( coc i rc d ) ( x ) = 4 x 2 + 20 x − 2 .
The final answer is 4 x 2 + 20 x − 2 .
Explanation
Understanding the Composition of Functions We are given two functions, c ( x ) = 4 x − 2 and d ( x ) = x 2 + 5 x . We want to find the composition ( coc i rc ) d ) ( x ) = c ( d ( x )) . This means we need to substitute the function d ( x ) into the function c ( x ) .
Substituting d(x) into c(x) To find ( coc i rc ) d ) ( x ) , we need to evaluate c ( d ( x )) . This means we replace every instance of x in the expression for c ( x ) with the entire expression for d ( x ) . So we have c ( d ( x )) = c ( x 2 + 5 x ) = 4 ( x 2 + 5 x ) − 2
Expanding and Simplifying Now we expand and simplify the expression: 4 ( x 2 + 5 x ) − 2 = 4 x 2 + 20 x − 2
Final Answer Therefore, ( coc i rc ) d ) ( x ) = 4 x 2 + 20 x − 2 .
Examples
Function composition is a fundamental concept in mathematics and has many real-world applications. For example, consider a store that marks up the price of an item by 20% and then applies a coupon for 5 o ff . I f x i s t h eor i g ina lp r i ce , t h e ma r k u p f u n c t i o ni s m(x) = 1.20x an d t h eco u p o n f u n c t i o ni s c(x) = x - 5 . A ppl y in g t h e ma r k u p an d t h e n t h eco u p o ni s t h eco m p os i t i o n c(m(x)) = 1.20x - 5$. This shows how function composition can model sequential operations in business and economics.
The composition ( c ∘ d ) ( x ) is calculated by substituting d ( x ) into c ( x ) , resulting in 4 x 2 + 20 x − 2 , which matches option D. Therefore, the answer is D. 4x^2 + 20x - 2.
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