Inverse functions....one part that I struggle to explain with this topic is why you only switch the variables when writing the equations for a function/relation, but not in a word problem. For example look at this site, http://www.classzone.com/eservices/home/pdf/student/LA207DBD.pdf If you go to that site....in example 1 for finding the inverse, you switch x and y. I get this---the domain and range switch so you flip the x and the y. How do you explain to kids that you do not switch the variables for word problems. For example, #3 on that link. For finding the inverse model, they simply solve the equation for w...they do not switch l and w. I mean, I get the idea intuitively, but I struggle trying to explain it. I see the little box suggests that it would be confusing since the variables represent real-life quantities, but I suspect students would wonder why to switch variables sometimes then. Any suggestions?

To be honest, I think they make it unnecessarily confusing. I think I would honestly tell them that when it's "Inverse model", they're merely switching which variable they solve for, not finding the inverse of the original function. As to why, I'm not sure I can come up with an itelligent answer.

Thanks. I may end up just skipping these types of word problems, or rewording them to avoid the confusion. Thinking about how they are solved and what they are used for, it almost makes more sense to include them in an algebra 1 section on literal equations. Edit: I think I will actually present them with variables where the variables don't directly relate to the quantities as shown here so it is more closely related to the lesson: http://www.youtube.com/watch?v=Ee2rgggiOGA

First of all, I would teach the kids not to switch the x and y until the last step. Solve for x first. Then as the final step, tell them if you leave the function as is, it's equivalent to the original function. This is sometimes useful for solving word problems that give you y, instead of x to start. BUT then tell them if you do switch the x and y, then they have effectively turned the original function into what's called an inverse function. This leads into your word problem where they first solve for w, then demonstrate that using this variation of the original equation, you can simply solve the word problem without needing to switch the variable. If that is your textbook, it sucks.

They're switching x and y unnecessarily because leaving them confuses people who haven't quite understood what variables and functions are. Further, that they can be called whatever you want. For these individuals, y is always sits by itself and there's a bunch of stuff involving x on the other side. I'd argue this is terrible, because it reinforces the idea that y always implies a function and that x is always a variable. It's additionally terrible because it further obfuscates the difference between an object in a collection versus the collection itself. Because this same exact stupid convention can cause problems in higher level math (analysis, topology, etc), we often switch to a slightly different convention: f(x) = y h(y) = x and f^-1(y) = h(y) where f and h are functions, and x and y are the elements related by the function.

Thanks---though I don't suspect too many of my kids will be taking analysis or topology. (It's a lower track math class.) Does make sense though.

Sorry, I should probably have been more clear - I would completely reword the first exercise to avoid the issue of renaming variables. Once you name an object, the name should not change.