I have found that inverse functions can be both frustrating and confusing for many students so I use direct instruction and guided practice, I Do - We Do - You Do gradual release for this lesson. I want students to get comfortable with process of writing inverses in general first so I don't focus on restricted domains as we "set the stage", unless a student notices and questions whether or not an example inverse is a function.
I begin with two problems posted on the board as shown in my Educreations video below. I made the video for students to reference later if they need review.
After working the first problem for them while they copy my work, I ask my students to work through the second problem with me. There are a few who will resist copying the work, saying they understand it already or that they can remember how to do it. I've found however that by insisting they copy problems I avoid the need to re-teach and I reduce student frustration by giving them a handy reference and support tool since I'll ask students who have questions to share their copied problems and show me where they got lost. Before moving on to individual practice I give a brief definition of inverse functions and show my students the inverse function notation. I explain that like determining odd and even functions, you can verify inverse fucntions by confirming that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x
For this section students work individually to hone their skills and understanding of inverse functions. I distribute the worksheet and explain that they have twenty minutes to finish these problems and then they should prepare to share with the class. (MP1) While they're working I walk around offering encouragement and assistance as needed. For example if a student is struggling with where to begin I might prompt them with "How can you rewrite this function using y instead of f(x)?" followed by "Okay, now how do you switch the variables - what would that look like?" This is usually sufficient, although I occasionally have a student who needs additional direct instruction, which I provide if time permits otherwise this gets schedule for out-of-class. This link offers an additional challenge/extension for bright students. When twenty minutes have passed or when everyone is done I randomly select students to post their solutions and work on the front board. I have room for 3-4 students at once, so no individual is put on the spot as they work. After the first set of problems is posted I ask the class to review and critique the work. (MP3) As they make corrections they know they are also expected to explain what they're doing and why, but sometimes I have to prompt with a question like "How did you know to do it that way instead of the way it was posted?" or "What let you know that it wasn't done correctly?" This is also when I introduce the idea of restricting the domain to make an inverse "work" as a function, by asking students to imagine running a horizontal line up the original function like they use the vertical line test for functions. Most students remember the vertical line test from Algebra, so this is a relatively easy connection to make. We continue until we've reviewed all twelve problems and everyone has had an opportunity to make corrections to their own papers. I've found that immediate feedback on practice problems helps students cement their understanding.
To wrap up this lesson I ask each student to write a "how-to" in their own words. I explain that they can use functions as part of the "how to" but should focus on clear directions written for a classmate who is absent today. (MP6) These are today's "ticket out the door." I explain why I value this activity in my video.