# Day 45: Inverse Insight

In Precalculus today, we debriefed yesterday’s quiz.  One particular question was from the University of Michigan Math 105 course based on the Hughes Haslett textbook that I love.  They post their old exams which are full of really thoughtful application questions.  The quiz question asked was

• Ivana B. Green is experimenting with a mixture of liquid fertilizer for her garden. She began with two liters of mixture, containing 200 ml of fertilizer. As she adds fertilizer, the concentration of fertilizer in the mixture changes.
• Find a formula which represents the concentration, C = f(x), of fertilizer in the mixture when Ivana adds x ml of fertilizer. (Note: 1 liter = 1000 ml.)
• Find a formula for f-1(C).

The second question about the inverse was awesome!  It brought out misinterpretations of symbols (Math Practice: precision) as well as defining variables (i.e. x is originally defined as the amount of fertilizer added and C is the concentration of fertilizer).  My students are very adept at finding inverses by “switching x and y.”  In fact, I found out they do it mindlessly because the only time they are asked to find inverses is in the abstract math world where x and y are not defined specifically.  You can see in the photo above that the students switched variables, used x and y instead of x and C, were sloppy in their final notation f, etc.  However, we talked about their new equation with x as the independent variable in the inverse function…but they never re-defined x to now represent the concentration.  Through discussion about what the variables meant, many had the ah-ha moment that the variables mean something especially in an application problem.