CMI2015-B-01

Question

Let $\Sigma=\{a,b\}.$ Given a language $L\underline\subset \Sigma^{\ast}$ and a word $w\in\Sigma^{\ast}$, define the languages:
$$Extend(L,w) :=\{xw\:|\:x\in L\}$$ $$Shrink(L,w) :=\{x\:|\:xw\in L\}$$Show that if $L$ is regular, both $Extend(L,w)$ and $Shrink(L,w)$ are regular.

Answer 1