This is the explanation why $V$ is an $F[x]$-module:

[![enter image description here](https://i.stack.imgur.com/qfb3I.png)](https://i.stack.imgur.com/qfb3I.png)

[![enter image description here](https://i.stack.imgur.com/CrckG.png)](https://i.stack.imgur.com/CrckG.png)

[![enter image description here](https://i.stack.imgur.com/FWpYO.png)](https://i.stack.imgur.com/FWpYO.png)

The book said "The linear map $T$ will enable us to make $V$ into an $F[x]$-module " why this happened? 

Also, I did not understand the statement " $\circ$ denotes composition of functions (which make sense because the domain and codomain of $T$ are the same )", what is the relation between composition of functions, and domain & codomain? could anyone tell me please?

The book said that "The definition of the $F[x]$ action on $V$ is consistent with the given action of the field $F$ on the vector space $V$", but I can not see how, could anyone explain this for me please? 

Could anyone explain the page for me in a clearer and simpler way please?

thanks!!