Functional, Declarative, and Imperative Programming

There's not really any non-ambiguous, objective definition for these. Here is how I would define them:

Imperative - The focus is on what steps the computer should take rather than what the computer will do (ex. C, C++, Java).

Declarative - The focus is on what the computer should do rather than how it should do it (ex. SQL).

Functional - a subset of declarative languages that has heavy focus on recursion

In a nutshell:

An imperative language specfies a series of instructions that the computer executes in sequence (do this, then do that).

A declarative language declares a set of rules about what outputs should result from which inputs (eg. if you have A, then the result is B). An engine will apply these rules to inputs, and give an output.

A functional language declares a set of mathematical/logical functions which define how input is translated to output. eg. f(y) = y * y. it is a type of declarative language.

I think that your taxonomy is incorrect. There are two opposite types imperative and declarative. Functional is just a subtype of declarative. BTW, wikipedia states the same fact.