If $A=\{1,2,3,4\}$, then which of the following are functions from A to itself ?

For function every element of domain must have a unique image in co-domain 

$f_1=\left\{\right(1,2),(2,3),(3,4\left)\right\}$. Thus the element 4 of domain does not have an image. 

$f_2=\left\{\right(1,4),(2,3),(2,4),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4\left)\right\}$ The uniqueness is violated as the elements have not one but many images. 

$f_3=\left\{\right(2,1),(3,2),(3,1),(4,1),(4,2),(4,3\left)\right\}$ Not a function as in $f_3$ uniqueness is violated.

$f_4=\left\{\right(1,4),(2,3),(3,2),(4,1\left)\right\}$ $f_4$ is a function as each element has a unique image.