Functional dependencies with the same key? let's consider a table with

carID | hireDate | manufactory | model | custID | custName | outletNo | outletLoc


I want to evaluate all the functional dependencies to bring in first, second and then third normal form.


Functional dependencies

carID,hireDate -&gt; custID

Partial dependencies

carID-&gt;manufactory, model, outletNo**

Transitive dependencies

custID-&gt;custName
outletNo-&gt;outletLoc



Since a car is in a outlet only I have in the partial dependecies this:

carID-&gt;manufactory, model, outletNo**


However this leads to anomalies in insertion (imagine adding a car with no outlet), so should not that be like this?

carID-&gt;manufactory, model
carID-&gt;outletNo


But isn't this still an normalisation anomaly?
I think you're misjudging what an insertion anomaly is. It does make sense for the FD to be 

carID -&gt; manufactory, model, outletNo**


since this is just a way to say that the same carID always has the same manufactory, model and outletNo.

Now, if you leave aside for a moment custId and custName, and you decompose the relation this way:

R1(carID, manufactory, model, outletNo), R2(outletNo, location)


you see that this is a fine decomposition that meets 3NF and is free from insertion anomalies: infact you just cannot have, for example, the same carID in two distinct outlets, or the same outletNo with different location. 

You're right that adding a car with no outletNo would be no good since outletNo is a foreign key to R2, infact most DBs won't allow you to do such a thing. But this is a different story. Not providing values for mandatory fields is just wrong, and that of course doesn't mean the design of the database is not correct.