$($a$)$ Construct NFA's A and $B,$ such that $L(A)=L(a{b}^{**}{a}^{**}),$ and $L(B)=L(a^{**}{b}^{**}a).$

Then use these NFA's to construct an NFA $C,$ such that $L(C)=L(A)L(B).$

$($b$)$ Suppose $h$ is the homomorphism from $\{0,1,2\}$ to $\{a,b{\}}^{**}$ defined by $h(0)=a,h(1)=b,$ and

$h(2)=ba.$

i$.$ What is the string $h(21120)?$

ii$.$ What is the language $h(L(0+1^{**}2))?$

iii. Let $L=\{ba\}.$ What is the language $h^{-1}(L)?$