the extended transition function
/ delta-hat = the symbol δ with a hat /
The ordinary transition function δ tells you what one symbol does. But a real input is a whole string, and we want to ask 'starting from state q, what state do I end up in after reading this entire string?'. The extended transition function, usually written δ-hat (a δ with a hat) or δ*, answers exactly that: it takes a state and a whole string and returns the single state you reach after consuming all of it.
It is defined by stitching δ together one symbol at a time, by induction on the length of the string. The base case is the empty string: δ-hat(q, ε) = q, meaning 'read nothing, stay put'. The step case peels off the last symbol: to read a string w followed by a symbol a, first read w to reach some state r = δ-hat(q, w), then take one more ordinary step, δ-hat(q, wa) = δ(r, a). Apply this rule repeatedly and you simply follow the path symbol by symbol, which is exactly what running the machine does.
Why bother naming it? Because it lets us state the meaning of the whole machine cleanly. A string w is accepted precisely when δ-hat(q0, w) lies in F, and the language of the DFA is { w : δ-hat(q0, w) ∈ F }. Without δ-hat we would have to talk informally about 'the path'; with it, acceptance and the recognised language become one-line definitions you can prove things about.
On the 'ends in 1' DFA (start A), compute δ-hat(A, 101) step by step: δ-hat(A, ε) = A; δ-hat(A, 1) = δ(A,1) = B; δ-hat(A, 10) = δ(B,0) = A; δ-hat(A, 101) = δ(A,1) = B. Since B is in F, the string 101 is accepted.
δ-hat threads a whole string through δ; the base case is δ-hat(q, ε) = q.
δ-hat is not a new mechanism — it is just δ applied repeatedly, given a name. The hat distinguishes 'reads a string' from the plain δ that 'reads one symbol'.