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When we talk about functions, we're dealing with special relationships between two sets of numbers or objects. Function notation is a way to represent these relationships in a formal mathematical way, and it makes it easier for us to communicate and think about functions. For example, if we have a function named 'g', we show this by writing it as 'g(x)', where 'x' is the input into the function. The 'g' essentially stands for 'given x, the function g produces...'

In the problem you're solving, you've encountered a function that is written in a set of ordered pairs. This specific representation is a way to visually present the function's input and corresponding output. When you see a function like \( g = \{(-8,-1),(-6,0),(2,7),(5,0),(12,10)\} \), each pair inside the curly braces is an ordered pair where the first element is an input 'x' and the second is the output 'g(x)' for that function.

Ordered pairs are the building blocks of functions when they are defined as a set of points. An ordered pair, typically written in the form (x, y), consists of two elements where 'x' represents the horizontal position and 'y' represents the vertical position on a coordinate plane. The first element of the ordered pair corresponds to the domain while the second refers to the range.

In the exercise given, \(g:\{(-8,-1),(-6,0),(2,7),(5,0),(12,10)\}\), each pair, like (-8,-1), directly tells us which input 'x' corresponds to which output 'y'. It shows us that when the function 'g' takes in -8, it gives out -1. Understanding this is crucial when determining the domain and range of the function.

Discrete functions are those where the domain (input values) is a set of isolated points. This is different from continuous functions, where the domain includes an entire interval of numbers. The function denoted in the exercise, represented as a set of ordered pairs, is an example of a discrete function because it only outputs values at certain points that are specifically defined.

This means that for our function \(g\), there is not an output for every possible number we could plug in as 'x', but only for the 'x' values that we have in our set: \{-8, -6, 2, 5, 12\}. Consequently, when identifying the domain and range of a discrete function, we can simply list the unique x-values and y-values respectively, as our solution describes.

Elementary algebra brings in the concept of using symbols and letters to represent numbers and quantities in mathematical formulas and equations. This is the branch of mathematics that introduces us to the concepts of variables and constants. Understanding algebra is essential when dealing with functions, domain, and range.

The domain and range in your exercise are determined by applying algebraic reasoning to identify and list the unique values from the ordered pairs. The skill involved here includes recognizing that the domain consists of all the first elements of the ordered pairs (the 'x' values), and the range is made up of all the second elements (the 'y' values). Hence, the domain and range of the function \(g\) are arrived at with elementary algebra techniques.