# Write a mathematical expression for the function to generate daily costs;fixed running cost $60 per dayEach belt made $3 (max 50 per day)What is the domain and range of this function?What is the...

Write a mathematical expression for the function to generate daily costs;

fixed running cost $60 per day

Each belt made $3 (max 50 per day)

What is the domain and range of this function?

What is the inverse of the above function? State inverse domain and range.

Desribe what the inverse function means in terms of cost and the number of belts made. Under what circumstances might the belt maker require such a function?

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### 1 Answer

`f(x) = 3x + 60`

Domain `{x in NN|xlt=50}`or `{0,1,2,...,50}`

Range `{y in NN|x in NN, x<=50,y=3x+60}`or `{60,63,66,69,...,207,210}`

Inverse:

(1)First replace f(x) with y to get

` y = 3x+60`

(2)Exchange x and y to get

` x = 3y + 60`

(3)Solve for y

`y = 1/3x-20`

`f^(-1)(x)=1/3x-20`

Domain is {60, 63, 66, 69,...,207,210}

Range is {0,1,2,...,50}

The inverse would allow us to target a given cost and find the number of belts we would have to make to generate that cost. So if you want a cost of 120, we could put it into the inverse function and find we have to make 20 belts.

The manufacturer could use the cost function along with a profit function to determine the optimum number of belts to make to maximize profit. They could also use this to determine the price they should sell an item at in order to optomize their profit.

So the answers are:

`f(x)=3x+60` Domain {0,1,...50}, Range {60,63,66,...,207,210}

`f^(-1)(x)=1/3x-20` Domain {60,63,66,...,207,210}, Range {0,1,2,...,49,50}