what is the difference between transition function (delta) and extended transition function (delta cap ) in finite automata
I read some references in order to answer your question ,

Transition function : takes as arguments a state and an input symbol and returns a state, denoted by δ .
Extended transition function : Describes what happens when we start in any state and follow any sequence of inputs ,means is a function that takes a state q and a string w and returns a state p (the state that the automaton reaches when starting in state q and processing the sequence of inputs w).  denoted by ˆδ .
in simple term : transition function takes two parameters: a state and a symbol , while extended transition function takes a state and a string.

Example of transition function :


Example of extended transition function :