If [tex]r(x)=3x-1[/tex] and [tex]s(x)=2x+1[/tex], which expression is equivalent to [tex](\frac{r}{s})(6)[/tex]?
A. [tex]\frac{3(6)-1}{2(6)+1}[/tex]
B. [tex]\frac{(6)}{2(6)+1}[/tex]
C. [tex]\frac{36-1}{26+1}[/tex]
D. [tex]\frac{(8)-1}{(6)+1}[/tex]
Answer (1)
Evaluate r ( 6 ) by substituting x = 6 into r ( x ) = 3 x − 1 , which gives r ( 6 ) = 3 ( 6 ) − 1 = 17 .
Evaluate s ( 6 ) by substituting x = 6 into s ( x ) = 2 x + 1 , which gives s ( 6 ) = 2 ( 6 ) + 1 = 13 .
Evaluate ( s r ) ( 6 ) = s ( 6 ) r ( 6 ) = 13 17 .
The expression equivalent to ( s r ) ( 6 ) is 2 ( 6 ) + 1 3 ( 6 ) − 1 .
Explanation
Understanding the Problem We are given two functions, r ( x ) = 3 x − 1 and s ( x ) = 2 x + 1 . We want to find an expression equivalent to ( s r ) ( 6 ) . Recall that ( s r ) ( x ) = s ( x ) r ( x ) .
Evaluating r(6) First, we need to evaluate r ( 6 ) . Substituting x = 6 into r ( x ) , we get r ( 6 ) = 3 ( 6 ) − 1 = 18 − 1 = 17 .
Evaluating s(6) Next, we need to evaluate s ( 6 ) . Substituting x = 6 into s ( x ) , we get s ( 6 ) = 2 ( 6 ) + 1 = 12 + 1 = 13 .
Evaluating (r/s)(6) Now, we can evaluate ( s r ) ( 6 ) = s ( 6 ) r ( 6 ) = 13 17 . We need to find which of the given options is equal to 13 17 .
Comparing with the Options Let's examine the given options:
Option 1: 2 ( 6 ) + 1 3 ( 6 ) − 1 = 12 + 1 18 − 1 = 13 17 . This matches our result. Option 2: 2 ( 6 ) + 1 ( 6 ) = 12 + 1 6 = 13 6 . This does not match our result. Option 3: 26 + 1 36 − 1 = 27 35 . This does not match our result. Option 4: ( 6 ) + 1 ( 8 ) − 1 = 7 7 = 1 . This does not match our result.
Final Answer Therefore, the expression equivalent to ( s r ) ( 6 ) is 2 ( 6 ) + 1 3 ( 6 ) − 1 .
Examples
Understanding function composition is crucial in many real-world applications. For example, in manufacturing, if r ( x ) represents the cost of producing x items and s ( x ) represents the revenue from selling x items, then ( s r ) ( x ) gives the cost-to-revenue ratio for x items. Evaluating this ratio at a specific value, like x = 6 , helps determine the financial efficiency of producing and selling 6 items. This kind of analysis is essential for making informed business decisions.
