Inverse Relationships in Rational Functions

This lesson explains how to determine the relationship between two numbers when their product is constant, focusing on simple rational functions. It reviews the concept of inverse proportionality, where one quantity increases as the other decreases, and emphasizes the importance of remembering that the product of two variables remains constant. A practical example using parking costs illustrates this inverse relationship. The lesson also covers common mistakes, such as forgetting the constant product, and explains why neither variable can be zero due to the zero product property. The lesson concludes with a summary of how to create equations for simple rational functions using graphs and tables.

What is the relationship between two numbers when their product is constant?

• They are directly proportional.

• They are unrelated.

• They are inversely proportional.

• They are equal.

What is a common mistake students make when dealing with simple rational functions?

• Assuming the variables are equal.

• Assuming the variables are directly proportional.

• Forgetting the product of the variables is constant.

• Forgetting the sum of the variables is constant.

In the parking fee example, what happens to the cost per person as the number of people increases?

• The cost per person decreases.

• The cost per person remains the same.

• The cost per person increases.

• The cost per person doubles.

How is the parking fee example represented graphically?

• Cost per person on both axes.

• A straight line graph.

• People on the y-axis and cost per person on the x-axis.

• People on the x-axis and cost per person on the y-axis.

What equation represents the inverse relationship in the parking fee example?

• p / m = 24

• p * m = 24

• p + m = 24

• p - m = 24

What is the constant product in the parking fee example?

• 24

• 12

• 48

• 36

What happens to the graph of a simple rational function as one variable increases?

• It curves downwards.

• It remains constant.

• It curves upwards.

• It becomes a straight line.

What does the zero product property imply in the context of simple rational functions?

• Either variable can be zero.

• Both variables must be zero.

• Neither variable can be zero.

• The product can be zero.

Why can't the number of people (p) or the cost per person (m) be zero in the parking fee example?

• Because the sum must remain constant at 24.

• Because the graph would be a straight line.

• Because the product must remain constant at 24.

• Because the total cost would be zero.

What is the main takeaway from the lesson on simple rational functions?

• Understanding the zero product property.

• Understanding direct proportionality.

• Understanding how to graph linear functions.

• Understanding inverse proportionality and constant product.