Question

1. function ( c ) is defined by the equation ( c(n) = 50 + 4n ). it gives the monthly cost, in dollars, of visiting a gym as a function of the number of visits, ( n ).

a. find the value of ( c(7) ). show your reasoning and explain what the value means in this situation.

( c(7) = )

explanation - what does this mean in this situation?

b. find the inverse function.

c. in this situation, what information would the inverse function give us?

Explanation:

Part a

Step1: Substitute \( n = 7 \) into \( c(n) \)

The function is \( c(n)=50 + 4n \). We substitute \( n = 7 \) into this function. So we have \( c(7)=50+4\times7 \).

Step2: Calculate the value

First, calculate \( 4\times7 = 28 \). Then, add 50 to 28: \( 50 + 28 = 78 \). In this situation, \( c(7) = 78 \) means that if you visit the gym 7 times in a month, the monthly cost will be 78 dollars.

Step1: Start with the original function

We have \( c(n)=50 + 4n \). Let \( y = c(n) \), so the equation becomes \( y=50 + 4n \).

Step2: Solve for \( n \) in terms of \( y \)

First, subtract 50 from both sides: \( y - 50=4n \). Then, divide both sides by 4: \( n=\frac{y - 50}{4} \).

Step3: Replace \( y \) with \( c^{-1}(n) \) and \( n \) with \( y \) (or just rewrite the inverse function)

The inverse function \( c^{-1}(n) \) (where \( n \) now represents the cost) is \( c^{-1}(n)=\frac{n - 50}{4} \).

The original function \( c(n) \) takes the number of visits \( n \) and gives the cost \( c(n) \). The inverse function \( c^{-1}(n) \) (where \( n \) is the cost) will take a given monthly cost and give the number of gym visits that would result in that cost. For example, if we know the total monthly cost, we can use the inverse function to find how many times we visited the gym.

Answer:

\( c(7)=78 \). This means the monthly cost for 7 gym visits is 78 dollars.

Part b