What is the importance of classifying function and relation?

Answer 1

These are applied in our everyday life! Not to mention in Physics, Engineering, Economics, Social Sciences, etc.

A very simple example. Suppose 1 gallon of a certain fuel you need costs 3 US$. So, there's a function that, to each volume V of this fuel, assigns the cost C in which you incur to by this volume. This function is given by C = 3 V. So, when you want to know how much you'll pay if you want the volume V, then you implicitly, probably even without realizing, you compute the value of this function. And also, again probably without realizing, you know this function is a bijection, that is, to each V there corresponds only one C and different values of V lead to different values of C.

Now suppose you have C dollars and you want to know how much fuel you can buy with these C dollars. Then you do a kind of inverse thinking, you compute V = C/3. What did you do, without realizing, or at least without thinking about? You determined the inverse of the function C = 3V. Computing V = C/3, you implicitly worked with the inverse of the previous function, that is, the function that, to each amount of money C, gives the amount V of fuel you can buy.

Isn't this a good reason to study functions and their inverses?