Below is a little Desmos application which is designed to assist in visualizing what the inverse of a function is. Two considerations are key when considering the inverse of a function:
- On what domain are we considering the function? Some functions may be invertible on part of their domain, but not all.
- When "reversing" the function (taking outputs back to inputs) do we have more than one possible answer? If yes, then there isn't an inverse function because the reverse process fails the requirement to become a function.
If you can draw a horizontal line through the graph of a function and only intersect once, no matter where you draw the line, then the function is invertible. This is called the "Horizontal Line Test". Finally, We say a function is one-to-one if each \(x\) maps to only one \(y\) and vice versa. One-to-one functions are automatically invertible.