Modeling with Inverses
In this activity, you will answer questions about the functions that model these three situations and their inverse functions.

Model 1: Scientists developed a process of producing fresh water from salt water. The amount of fresh water produced from salt water in x hours can be found using a logarithmic function (base e) of . In the function, the number 3 represents the amount of fresh water produced after 1 hour has passed and the number 2 represents the rate of production of fresh water.

Model 2: Jack owns a company that manufactures a product that we’ll call product A. After producing and selling x units of product A, the profit his company made is represented by the cube root function .

Model 3: A biologist has tracked the population of Antarctic penguins in a marine reserve since 2012. He represents the population with the rational function , where x is the number of years that have passed since 2012 and f(x) is the number of penguins.

Part A

Question 1
Question
Enter the correct answer in the box.
Write the inverse of the logarithmic function used to represent the situation in model 1, . Express the answer in radical form.
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Question 2
Question
Enter the correct answer in the box.
Write the inverse of the cube root function used to represent the situation in model 2, .
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Question 3
Question
Enter the correct answer in the box.
Write the inverse of the rational function used to represent the situation in model 3, .
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Part B
How does each inverse function from part A relate to its original function? Describe the relationship in terms of the context for each situation. Do this for each of the three models.

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Part C

Question 1
Question
Graph the function and its inverse from model 1.
-10-8-6-4-2108642-10-8-6-4-2108642x y
Question 2
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Graph the function and its inverse from model 2.
-10-8-6-4-2108642-10-8-6-4-2108642x y
Question 3
Question
Graph the function and its inverse from model 3.
-40-35-30-25-20-15-10-5403530252015 105-20-15-10-5403530252015105xy
Question 4
For each pair of graphs in part C, what relationship do you notice for f(x) and f-1(x)?

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Part D
Why might it be beneficial to have both an inverse function and the original function when attempting to model a situation? You may use a specific function type or scenario in your answer.

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