The function $c(n)$ below relates the number of bushels of apples picked at a pick-your-own orchard to the final cost for the apples.
It takes as input the number of bushels of apples picked after paying an entry fee to an orchard, and it returns as output the cost of the apples (in dollars).
$c(n)=10 n+20$
Which equation below represents the inverse function $n(c)$, which takes the cost of the apples as input and returns the number of bushels picked as output?
A. $n(c)=\frac{c-20}{10}$
B. $n(c)=\frac{c+20}{10}$
C. $n(c)=\frac{c+10}{20}$
D. $n(c)=\frac{c-10}{20}$
Answer (1)
Start with the given function: c ( n ) = 10 n + 20 .
Solve for n in terms of c : c − 20 = 10 n .
Divide by 10 to isolate n : n = 10 c − 20 .
The inverse function is: n ( c ) = 10 c − 20 .
Explanation
Understanding the Problem We are given the function c ( n ) = 10 n + 20 , which represents the cost c of picking n bushels of apples. We want to find the inverse function n ( c ) , which represents the number of bushels n picked for a given cost c .
Finding the Inverse Function To find the inverse function, we need to solve the equation c = 10 n + 20 for n in terms of c .
Isolating the Term with n First, subtract 20 from both sides of the equation: c − 20 = 10 n
Solving for n Next, divide both sides of the equation by 10: n = 10 c − 20
The Inverse Function Therefore, the inverse function is n ( c ) = 10 c − 20 .
Final Answer Comparing this to the given options, we see that option A matches our result.
Examples
Imagine you're at the orchard and want to figure out how many bushels of apples you can pick with a certain amount of money. Knowing the inverse function helps you determine exactly that! For example, if you have 70 , yo u c an c a l c u l a t e t h e n u mb ero f b u s h e l syo u c an p i c k : n(70) = \frac{70 - 20}{10} = \frac{50}{10} = 5$ bushels. This is a practical application of inverse functions in everyday scenarios.
