In elementary school, students learn how some mathematical operations are opposites of other operations. Addition is the opposite of subtraction. Multiplication is the opposite of division. As students get older, we introduce the word inverse. Inverse operations undo what has been done and are essential for solving equations. In algebra we expand the idea of inverses to functions. In geometry, the inverse of a conditional statement is when the hypothesis and conclusion are negated. In calculus students learn that integration is the inverse of taking the derivative.
Today we are going to delve into inverses with respect to the algebraic lens of functions, both analytically and with the geometric connection graphically.
Graphical Inverse
A function and its inverses are connected graphically by being the reflection over the line y=x.
When I start teaching inverses of functions, I often will start with points on the coordinate plane and have students switch the x and y coordinates to find the inverse. If you do this with several points and then ask students to graph the line y=x, they often will notice the connection of the reflection over the line y=x on their own. Here is a sample prompt to give your students.
We would discuss the domain and range of the function and its inverse as well as deciding if the inverse is still a function or just a relation.
Next, we look at a linear function, because this is the function that my students are most comfortable with and you do not need to restrict the domain for the inverse to be a function.
I may use a function similar to the graph shown in the first image above, f(x)=2x+4. Students will table the function and then switch the x and y coordinates to graph the inverse. They then would graph the line y=x and really see the connection that the function and its inverse are reflected over the line y=x. It would be good to note whether the inverse of the function is also a function at this point, as it will set up a future discussion when graphing parabolas.
After students start to get a feel for what an inverse looks like graphically, it is time to introduce the algorithm for finding the inverse.
Steps to Find the Inverse of a Function Algebraically
We will go over the steps and an example followed by some practice problems.
Restricted Domains
Finally, we will look at functions that need to have a restricted domain to make the function’s inverse also a function. It should be noted that if the inverse is not a function, then we would classify it simply as a relation.
The parent function for quadratics is a great function to start with when looking at restricted domains.
How do you teach inverse functions? Leave a comment below! If you are interested in my lesson or more teaching resources for inverse functions, visit my TPT store below!
