Understand why a function must be one-to-one, or invertible, in order for the inverse mapping to be a function.
Explore relationships between the graph of a function and its inverse, including their domains and ranges.
Quick Lesson Plan
Today students continue reasoning about inverse functions but from a more analytical perspective. The goal is to answer the question: “how can we tell if a function’s inverse will also be a function?”
Students graph the elevation of a hiker over time, keeping in mind that this is a rough sketch since we only have a few data points. Students should articulate in question 2 that elevation is a function of time because each time is associated with only one elevation; a hiker can’t be in two places at once. While we expect many students to simply state the elevation graph passes the vertical line test, push students to really explain what this means.
Question 3 has students thinking contextually about one-to-one functions and how this elevation function does not meet such requirements. This ties into the idea of question 4c, namely that there are two times where any given elevation (except the summit) is reached. In question 5, this knowledge is expanded to include a graphical interpretation: the horizontal line test.
In question 5, it is important that students realize why having repeated elevations is problematic, not for the original, but for the inverse. For groups that are struggling to see this, have students identify the two ordered pairs on the original graph that produce a given elevation, and then ask what ordered pairs would be on the inverse. This line of questioning illuminates how the same input will go to multiple outputs, which contradicts the definition of a function.
When the x-value of the inverse relation is 12,978, what would the output be?
How can we determine whether a relationship represents a function or not? What if we don’t have a graph?
Why are repeated outputs on the original function problematic?
Can you picture what the inverse graph would look like?
The most challenging aspect for students here is using precise communication. Vague language like “it shows whether it’s a function or not” is misleading because we now have two different tests and two different functions. Make sure that students can communicate that if the original function fails the horizontal line test, it is the inverse relation that fails to be a function. Some students overgeneralize here and conclude that a function has to be one-to-one to even be a function, when in fact this is only a requirement for the inverse to be a function.
We purposefully do not have students sketch an inverse graph during the experience because we want them to be able to determine if the inverse is a function from the original function. Certainly students know that passing the vertical line test guarantees a function, inverse or not, but the point is that we don’t need to actually graph the inverse to find this information. We add the sketch of the inverse in the debrief to support students’ conclusion from the horizontal line test.