When a function always produces the same output for the same input and has no observable side effects, what key functional programming principle does it embody?

The key functional programming principle embodied is pure functions. A pure function is characterized by two fundamental properties, both directly described in the question. First, it is deterministic, meaning that given the exact same input arguments, it will consistently produce the exact same output every single time it is invoked. Its result depends solely on its inputs, never on any hidden or external state that could change between calls. Second, a pure function causes no observable side effects. A side effect is any interaction a function has with the outside world beyond returning a value. This includes, but is not limited to, modifying global variables, performing input or output operations like writing to a console or a file, making network requests, or mutating its input arguments. The absence of side effects ensures that calling a pure function does not alter the program's state in any way that can be observed by or affect other parts of the system, apart from the value it returns. For instance, a function `multiply(a, b)` that calculates `a b` is pure because it always yields the same product for the same `a` and `b`, and it does nothing else like printing to the screen or changing a global counter.