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Church's Thesis, often regarded as the cornerstone of theoretical computer science, posits that a function can be considered effectively computable if there exists an algorithm that can compute it. This notion intertwines the intuitive idea of what it means for a function to be calculable with an algorithm and the more formal mathematical framework of computability theory.

An important consequence of Church's Thesis is that it allows us to equate the terms 'effectively computable' and 'algorithmically computable' across various computational models. Although it is not a theorem that can be formally proven, Church's Thesis has withstood the test of time through empirical validation. Numerous models of computation, such as Turing Machines, Lambda Calculus, and Unrestricted Register Machines (URM), have all been shown to be equivalent in terms of their computational power, which supports the thesis.

Effective computability is a concept that involves determining whether or not a function can be computed within a finite number of steps by an algorithm or a mechanical process. A function that is effectively computable lends itself to a procedure that is not only deterministic but also doesn't require any form of ingenuity or creative insight to execute.

In the context of our exercise, both functions f(x) and g(x) are said to be effectively computable, suggesting that for any input within their domains, there is a known method to determine the corresponding outputs logically, systematically, and conclusively. This is the essence of what it means for a function to be effectively computable and is critical for proving the computability of the function h(x).

A function's domain is a foundational concept in both mathematics and computer science, referring to the set of all possible inputs over which the function is defined. In the realm of computable functions, understanding the domain is crucial because it dictates which inputs the function can process, and consequently, which outputs it can produce.

In practical terms, when examining computable functions such as f(x) and g(x) in our example, their domains signify the specific values that we can feed to their corresponding algorithms to receive an output. It's necessary to understand the domains of these functions to resolve the computability of h(x), which is predicated on the intersection of their domains, Dom(f) ∩ Dom(g). Only for inputs falling within this intersection does the function h(x) produce an output, emphasizing the function's reliance on the well-defined domains of f and g.

Recursive function theory, a branch of mathematical logic and computability theory, deals with the study of recursive functions—those that can be defined by a set of rules that apply to smaller or simpler instances of the same problem. This theory provides a framework for understanding which functions are computable and how they can be computed.

In the scope of URM-computable functions, recursive function theory helps formalize the relationship between algorithms and functions. It allows us to categorize functions based on their complexity and the methods required for their computation. Given that f(x) and g(x) are effectively computable, we infer they are also recursive. Therefore, h(x), being defined in terms of operations on f(x) and g(x), must also adhere to the principles of recursive function theory, supporting the claim that it is URM-computable.