If [tex]f(n)=2 n^2+2 n[/tex] and [tex]g(n)=n+1[/tex], what is [tex](f \div g)(n)[/tex] when [tex]n =-2[/tex]?
Answer (1)
First, express ( f ÷ g ) ( n ) as g ( n ) f ( n ) = n + 1 2 n 2 + 2 n .
Then, factor the numerator: 2 n 2 + 2 n = 2 n ( n + 1 ) .
Simplify the expression by canceling the common factor: n + 1 2 n ( n + 1 ) = 2 n .
Finally, substitute n = − 2 into the simplified expression: 2 ( − 2 ) = − 4 , so the answer is − 4 .
Explanation
Understanding the Problem We are given two functions, f ( n ) = 2 n 2 + 2 n and g ( n ) = n + 1 . Our goal is to find the value of ( f o b re ak ÷ g ) ( n ) when n = − 2 . The expression ( f o b re ak ÷ g ) ( n ) means g ( n ) f ( n ) .
Setting up the Division First, let's find the expression for g ( n ) f ( n ) . We have g ( n ) f ( n ) = n + 1 2 n 2 + 2 n .
Factoring the Numerator Next, we can factor the numerator of the expression: 2 n 2 + 2 n = 2 n ( n + 1 ) .
Simplifying the Expression Now, we can simplify the expression by canceling the common factor of ( n + 1 ) from the numerator and the denominator: n + 1 2 n ( n + 1 ) = 2 n , provided that n e q − 1 .
Substituting n = -2 Since we want to evaluate the expression at n = − 2 , the simplified expression 2 n is valid because − 2 e q − 1 . Now, we substitute n = − 2 into the simplified expression: 2 n = 2 ( − 2 ) = − 4.
Final Answer Therefore, ( f o b re ak ÷ g ) ( n ) when n = − 2 is − 4 .
Examples
Understanding function division is useful in many real-world applications. For example, if f ( n ) represents the total cost of producing n items and g ( n ) represents the number of items produced, then ( f ÷ g ) ( n ) gives the average cost per item. If you know how the total cost and number of items change with n , you can determine the average cost for a specific number of items. This is crucial for businesses to understand their cost structure and make informed decisions about pricing and production levels.
