Given [tex]f(x)=3 x+1[/tex] and [tex]g(x)=x^3-8[/tex], find [tex](f \cdot g)(x)[/tex].
A. [tex]3 x^3+x^2-6 x-6[/tex]
B. [tex]4 x^2-6 x-8[/tex]
C. [tex]3 x^3+x^2-18 x-6[/tex]
D. [tex]3 x^2-17[/tex]
Answer (2)
Find the composition f ( g ( x )) by substituting g ( x ) into f ( x ) .
Substitute x 3 − 8 into f ( x ) = 3 x + 1 , which gives f ( g ( x )) = 3 ( x 3 − 8 ) + 1 .
Simplify the expression: 3 x 3 − 24 + 1 = 3 x 3 − 23 .
The final answer is 3 x 3 − 23 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 3 x + 1 and g ( x ) = x 3 − 8 . The problem asks us to find ( f , 8 ) ( x ) . However, the notation ( f , 8 ) ( x ) is not standard. I assume that it means to find the composition f ( g ( x )) .
Composition of Functions To find f ( g ( x )) , we need to substitute g ( x ) into f ( x ) . This means we replace the x in f ( x ) with the entire expression for g ( x ) . So, we have
f ( g ( x )) = f ( x 3 − 8 )
Substitution Now, we substitute x 3 − 8 into f ( x ) = 3 x + 1 :
f ( g ( x )) = 3 ( x 3 − 8 ) + 1
Simplification Next, we simplify the expression:
f ( g ( x )) = 3 x 3 − 24 + 1
f ( g ( x )) = 3 x 3 − 23
Final Answer Therefore, ( f , 8 ) ( x ) = f ( g ( x )) = 3 x 3 − 23 . Comparing this result with the given options, we see that none of the options match our answer. However, there might be a typo in the question or the options. If the question was to find f ( g ( x )) , then the correct answer would be 3 x 3 − 23 .
Examples
Function composition is a fundamental concept in mathematics and has numerous real-world applications. For example, consider a scenario where a store offers a discount of 20% on all items, and there's an additional coupon for 10 o ff t h e d i sco u n t e d p r i ce . I f f(x) = 0.8x re p rese n t s t h e 20 g(x) = x - 10 re p rese n t s t h eco u p o n , t h e n f(g(x))$ would calculate the final price after applying both the discount and the coupon. Understanding function composition helps in optimizing processes and modeling complex relationships in various fields like economics, engineering, and computer science.
The product of the functions f ( x ) = 3 x + 1 and g ( x ) = x 3 − 8 results in 3 x 4 + x 3 − 24 x − 8 . This expression does not match any of the given options. The approach demonstrates the use of distributive property in multiplying polynomials.
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