Every input in the domain must be used but not everyone in the domain
When we say f:A->B, the codomain must not be all ———-
[Diagram showing that]
The part of the codomain that is actually hit is the range or image of the Function example x^
That image is always a subset of the codomain
Now the preimage operation is a mapping that takes a subset of the codomain and map it back to the domain. If the domain exists it maps back to the original. If there isn’t corresponding element it gives us the empty
Let’s explore this function right here. What if you give it the image?
Let’s take our x^2 defined for f:R->R
Sometimes preimage gives use something and sometimes if gives us the empty set. The preimage can’t be a function based on the properties we evoked earlier
The preimage operation gives us a set
It always exists since we haven’t even forced him to be a function