Injectivity

A function is Injective if, and only if, no two input variables have the same output.

But what does this mean? Why is it important?

Consider the following functions:and, let's take the following variables as input, the set f(x) will transform this set to the following set while g(x) will transform this set x to the following set

Let's say that you want to somehow find what you had before applying that function. For f(x), easy, we can get back our set x by just subtracting 3.

But, for our set, g(x), it seems that the same outputs 1 and 4 correspond to more than 1 input. How do we know which one is which? We can't. We can't undo this function, even if we simply just took the square root, because there's no guarantee we'll get back the original set.

As you probably have figured out by now, a function that is injective is invertible.

That is what we'll explain next.
The Inverse of a Function