1.9 Inverse functions

This is definitely two days, possibly three.

Just as with composite functions, it is useful to look at this three different ways: in terms of the function game, in terms of real world application, and in terms of the formalism.

• 1 Ask a student to triple every number you give him and then add 5. Do a few numbers. Then ask another student to reverse what the first student is doing . This is very easy. You give the first student a 2, and he gives you an 11. Then you give the second student an 11, and he gives you a 2. Do this a few times until everyone is comfortable with what is going on. Then ask what function the second student is doing. With a little time, everyone should be able to figure this out—he is reversing what the first student did, so he is subtracting five, then dividing by 3. These two students are “inverses” of each other—they will always reverse what the other one does.
• 2 Give a few easy functions where people can figure out the inverse. The inverse of $x+2$ is $x-2$ (and vice-versa: it is always symmetrical). The inverse of $x^3$ is $\sqrt[3]{x}$

The key thing to stress is how you test an inverse function . You try a number. For instance…

The point is that you take any number and put it into the first function; put the answer in the second function, and you should get back to your original number. Testing inverses in this way is more important than finding them, because it shows that you know what an inverse function means .

• 3 Ask for the inverse of $x^2$ . Trick question: it doesn’t have any! Why not? Because $x^2$ turns 3 into 9, and it also turns –3 into 9. It’s allowed to do that, it’s still a function. But an inverse would therefore have to turn 9 into both 3 and –3, and a function is not allowed to do that—rule of consistency. So $x^2$ is a function with no inverse. See if the class can come up with others. (Some include $y=\left|x\right|$ and $y=3$ .)
• 4 Now ask them for the inverse of $10-x$ . They will guess $10+x$ or $x-10$ ; make sure they test! They have to discover for themselves that these don’t work. The answer is $10-x$ ; it is its own inverse. (It turns 7 into 3, and 3 into 7.) Ask for other functions that are their own inverses, see if they can think of any. (Other examples include $y=x$ , $y=-x$ , $y=20/x$ .)
• 5 In practice, inverse functions are used to go backwards, as you might expect. If we have a function that tells us “If you work this many hours, you will get this much money,” the inverse function tells us “If you want to make this much money, you have to work this many hours.” It reverses the $x$ and the $y$ , the dependent and independent variables. Have the class come up with a couple of examples.
• 6 Formally, an inverse function is written $f^{-1}\left(x\right)$ . This does not mean it is an exponent, it is just the way you write “inverse function.” The strict definition is that $f\left(f^{-1}\right(x\left)\right)=x$ . This definition utilizes a composite function! It says that if $x$ goes into the inverse function, and then the original function, what comes out is… $x$ . This is a hard concept that requires some talking through.