This section introduces the geometric viewpoint of invertability.
A recurring perspective as we move toward studying individual functions types will be the idea of inverting a function. Remember that a function is a relationship between some domain and a codomain, where it “maps” each domain point to a (single) point in the codomain.
Inverting a function is “merely” the process of reversing the direction of . We will denote the inverse function by and we can see below what this looks like in terms of our domain and codomain.
There are a few subtle and key observations that can be made from this seemingly simple diagram however. The most obvious, observation we can make is that the codomain of the original function becomes the domain of the inverse function, and the domain of the original function becomes the codomain of the inverse function. That is to say; the role of domain and codomain switch for the inverse function.
This is a lot more important than it might initially seem, for two reasons. First, the inverse function taking an entire codomain as it’s domain could be rather problematic. Take, for example, the function defined by . The inverse function for would be (you can just take this on faith for now, we’ll cover this later). But if we try to use the entire codomain (ie ) as the domain for the inverse, then we would have a problem because the domain of is not . It turns out though that the range of is actually , not .
So, it is more helpful to take the range of the function as the domain of it’s inverse rather than the codomain. With this adjustment our picture would look like:
By using the range of as the domain of , we make sure that every point in the domain of is defined. Another way to say this is that we only consider the ‘points that actually came from some -value’ when we reverse the relationship to make the inverse relation.
Here is a video with more!