quanta'namo nay!

You have already specified that you want a specific output (0). If I translate this into words of spoken language you get: What input gives you an output of zero when squared and subtracted four? The deterministic answer is of course +-2. Determinism ensures that your answer to this question is always the same. If something is random then the frequencies of occurrence (probabilities) are also random and not constant.

In the sense of repeatability yes. Always when you input the same input to the same function it will deterministically give you the same result. Assuming no errors were made

A tricky question would be how does one define the square root of negative infinity

Probabilistic calculations are also essentially deterministic in their mathematical form. If virtual particles were an accurate description of reality they would also need to be deterministic as they are said to form the subatomic processes, given that the operation is not random and unpredictable. I'd say that if something is random it can not follow any process that can be modelled (i.e. is predictable). I suppose the inputs to a mathematical equation must be other equations as a substitutions or numeric values. Mathematics as the language only deals with equations and numeric values, physics as an application of that language then adds concepts like types (e.g. cats and dogs). An equation (or function) may have multiple inputs and the output may be an equation or a numeric value. If you use sets then you are talking about multiple instances of usage of the function I suppose. If you have a different view point to this I'd ask you to present a simple example with some simple set(s).

x is the input. Any given input will deterministically produce the output. Just that your equation will reduce to 0 = 0 and takes no input...

What I meant was that any mathematical equation f(x) when given input X and resulting in output Y, will always result in output Y with the input X. In that sense all mathematical equations are deterministic. If they were not, no mathematical proof could be made at all. I suppose the requirement of mathematical proofs is deterministic equations. An equation may have multiple solutions, but they're not random but deterministic as they always hold.

I've recently stumbled on some people claiming that some mathematical equations can produce random results. To me this seemed quite strange. Essentially, to my understanding mathematical equations are always by definition deterministic and this results in the possibility to make predictions in terms of science. Granted, my understanding often needs revising and I enjoy doing that upon encountering solid arguments. That means I am learning and my mind is evolving. It seems to me that if science is to use mathematics to approximate the behaviour of the Universe and if that modelling is successful, that in turn indicates that the Universe is entirely deterministic. There is of course the notion of quantum mechanics and "randomly existing virtual particles". One question that arises here is how could something that is random be possibly modelled by entirely deterministic means. However, I think the main question here I want to present to all mathematicians: How would you respond to someone claiming that a mathematical equation can produce random results? And additionally: Would you consider successful mathematical modelling of the universe as indication of the universe being deterministic?