TOC Extended Transaction Function of DFA

Question

δ*(q,cya) is equivalent to
(a) δ((q, y), a) (b) δ(δ*(q, y), a)
(c) δ(q, ya) (d) independent from δ notation
Solution: Option (b)

i have confusion that if getting input c on q  goes a another state x.

than ho can b option correct.

Comments

i think it is incorrect...

what if transition is δ*(q,c), then according to solution given above it will be equal to δ(δ*(q, ϵ), ϵ)...and in DFA with δ(δ*(q, ϵ), ϵ), it will be in same state 'q' , but δ*(q,c) can take to some other state...