Lesson 5 Flipped Out Develop Understanding

Learning Focus

Model a context using two different ways of thinking about the variables.

What are characteristics of inverse functions?

Open Up the Math: Launch, Explore, Discuss

Chandler and Isaac both like to ride bikes for exercise. They are discussing whether or not they have a similar pace so that they can plan a time to bike together when they find that they both think about bike riding in a different way. Chandler says, “Sometimes I have a busy schedule and it’s hard to fit in a ride. I think about how much time I have to ride and then how many miles I can go.” Isaac says, “As I’m training for a race, I think about how many miles I need to ride and then how much time it will take.” They both look at each other blankly and decide to do a little mathematical modeling.

Chandler says, “If I have $60$ minutes to ride, I can go $12$ miles.”

1.

Model Chandler’s way of thinking about her ride using a table, graph, equation, and any other representation that you think demonstrates Chandler’s method. Be sure to label each representation.

2.

Explain the connections between the representations you found.

3.

How many miles will Chandler travel at $30$ minutes? What is Chandler’s speed?

Isaac explains that he plans his ride based on the number of miles he needs for his training regimen. He knows that for every $5$ miles he needs to go, he plans $25$ minutes.

4.

Model Isaac’s way of thinking about a ride, with miles as the input, using a table, graph, equation, and any other representation that you think demonstrates his method. Be sure to label each representation.

5.

What is Isaac’s speed? How is this different from how Chandler describes her speed? Who is faster?

6.

How many miles will Isaac travel at $30$ minutes?

7.

Using the equations, tables, and graphs for Isaac and Chandler, make a list of observations about how these situations compare to each other.

Ready for More?

Using relationships that you have observed in this task, find the inverse of $f (x) = 3 x + 2$ . Justify your solution using multiple representations.

Takeaways

A function and its inverse:

Lesson Summary

In this lesson, we learned about inverse functions using a context that had two different and useful ways to think of the relationship. Two functions are called inverse functions when their inputs and outputs have been switched.

Retrieval

Solve for $y$ .

1.

$8 + x y = 14 x - 2 y$

Rewrite each expression.

2.

$3 \sqrt{7} (4 + \sqrt{7})$

3.

$(2 - 5 \sqrt{2}) (3 + 4 \sqrt{2})$