If [tex]$f(n)=2 n^2+2$[/tex] and [tex]$g(n)=n+1$[/tex], what is [tex]$(f-g)(n)$[/tex] when [tex]$n =-1$[/tex]?

First, find the expression for ( f − g ) ( n ) by subtracting g ( n ) from f ( n ) : ( f − g ) ( n ) = f ( n ) − g ( n ) .
Substitute the expressions for f ( n ) and g ( n ) : ( f − g ) ( n ) = ( 2 n 2 + 2 ) − ( n + 1 ) .
Simplify the expression: ( f − g ) ( n ) = 2 n 2 − n + 1 .
Substitute n = − 1 into the simplified expression and calculate the value: ( f − g ) ( − 1 ) = 2 ( − 1 ) 2 − ( − 1 ) + 1 = 4 . The final answer is 4 ​ .

Explanation

Understanding the Problem We are given two functions, f ( n ) = 2 n 2 + 2 and g ( n ) = n + 1 . We want to find the value of ( f − g ) ( n ) when n = − 1 . This means we need to first find the expression for ( f − g ) ( n ) and then substitute n = − 1 into that expression.

Finding the Expression for (f-g)(n) First, we find the expression for ( f − g ) ( n ) by subtracting g ( n ) from f ( n ) : ( f − g ) ( n ) = f ( n ) − g ( n ) Substituting the given expressions for f ( n ) and g ( n ) , we get: ( f − g ) ( n ) = ( 2 n 2 + 2 ) − ( n + 1 ) Now, we simplify the expression: ( f − g ) ( n ) = 2 n 2 + 2 − n − 1 = 2 n 2 − n + 1

Substituting n = -1 and Calculating the Value Next, we substitute n = − 1 into the simplified expression: ( f − g ) ( − 1 ) = 2 ( − 1 ) 2 − ( − 1 ) + 1 Now, we calculate the value: ( f − g ) ( − 1 ) = 2 ( 1 ) + 1 + 1 = 2 + 1 + 1 = 4

Final Answer Therefore, ( f − g ) ( n ) when n = − 1 is equal to 4.


Examples
Understanding function operations like ( f − g ) ( n ) is useful in many real-world applications. For example, imagine f ( n ) represents the total revenue of a company based on the number of products sold, n , and g ( n ) represents the cost of producing those n products. Then, ( f − g ) ( n ) would represent the profit of the company. If you want to know the profit when n = − 1 , it might represent a scenario where the company is operating at a loss or is in a debt situation, and calculating ( f − g ) ( − 1 ) would give you the value of that loss or debt.

Answered by GinnyAnswer | 2025-07-05