# Section 4.4: Analysis

Hungerford starts this section with the idea of induced function. The idea behind this is that there may be multiple functions that map to the same locations in a codomain but the functions themselves may not be unique.

Yet possible the most interesting portion of this section is the remainder theorem. It provides a fast way to calculate the remainder of a polynomial function when dividing the function by $x-a$ where $a\in F$. Applications of this would be interesting to delve into. This could be used in doing tasks like keeping stock of specific items etc.