Let's take Σ={a, b} and Γ={0, 1}.
Now let h be a homomorphism function where hΣ=>Γ*
Let's take an example to better understand this.
Let L={aa, ba, b} be a finite regular language now given h(a)=00, h(b)=10
Now we have to find h(L)?
Just substitute 00 in the place of a whereever in L and 10 in place of b.
We get, h(L)={0000, 0100, 10}
See that |L| =3 and |h(L)| =3, therefore |L|=|h(L)| in this example but this is not always true. Sometime there may arrive some duplicates h(L) while substution but we can guarantee that |h(L)| <= L
Let's take an example for inverse homomorphism as the name suggests we have to do reverse of homomorphism
Let L={0101} be a finite regular language now given h(a)=0, h(b)=1, h(c)=01
Now we have to find h-1(L)?
Just substitute a in the place of 0 whereever in L and b in place of 1 and c in place of 01.
We get, h-1(L)={abab, abc, cab, cc}
See that there are total 4 possible combinations and |L| =1 and |h-1(L)| =4, therefore we cannot guarantee that |h(L)| <= |L| or |h(L)| >= |L|