An Algebraic and Graphical Approach to Inverse Functions

This lesson outline allows the teacher to guide students through 2 very different ways to think about inverse functions. I usually begin with the algebraic approach because it is more challenging to students, and they seem more willing to strain their brains at the start of a lesson.
 
In the first four examples, students get a chance to determine whether the functions are inverses or not by determining if their compositions are equivalent.
 
After that, the students are introduced to the algorithm that enables them to find the "potential" inverse function of any given function. I say, "potential" because sometimes what seems to be the inverse of a function isn't a function at all; it's a relation.
 
The next part of the lesson involves graphing functions and their inverses simply by reversing the domain and range for each point. It becomes even clearer when they graph the functions and inverses in two colors. At this point, I have them sketch in the line of symmetry just by looking at the lines/curves they graphed, but I stop short of identifying the equation of the symmetry line.
 
All of this leads up to graphing a few more pairs of functions and inverses using Desmos. Even though the students are required to have and use TI-84 graphing calculators in my school, working with Desmos gives them a much clearer picture of the relationship between functions, inverses and their line of symmetry.
 
Here is the lesson outline. Go ahead and tailor it to your students' needs.