If you’re sufficiently geeky, you will surely want to know something unexpected about the mathematics of functions and their inverses: cryptography in the 15th Century. Why? Because then we’re focusing on the transition from the monoalphabetic ciphers (such as Caesar, affine, and other simple substitution ciphers) to the polyalphabetic ones (such as Vigenère, matrix, and so forth). Check out this post by Nick Pelling for more info than you probably want. In particular, look at the 1401 cipher table reproduced above. As Pelling points outs, at first glance it appears ridiculously insecure, since the ciphertext alphabet is simply the plaintext alphabet reversed. Except…each vowel has multiple ciphertext equivalents. Before you complain that this means that the cipher isn’t a function, notice that its inverse is a function, and that’s what’s important: the legitimate recipient of a message can decipher it readily for that reason. But frequency analysis will be radically thrown off, possibly foiling or at least delaying the cracker. There’s something deep here that will be useful in Algebra II.