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Understanding the domain of a function is fundamental in mathematics. The domain of a function refers to all possible input values (usually denoted by \(x\)) that will not lead to a mathematical error when input into the function. Examples of such errors include division by zero or taking square roots of negative numbers.

For instance, with the function \(g(x) = \frac{1}{x^2}\), the domain is all real numbers except \(x = 0\) because division by zero is undefined in mathematics. We represent the domain of \(g(x)\) with the interval notation \((-\infty, 0) \cup (0, \infty)\).

Similarly, when finding the domain of compositions like \((f \circ g)(x)\), it's crucial to consider the individual domains of both \(f(x)\) and \(g(x)\). In this case, due to the domain restriction of \(g(x)\), the domain of \((f \circ g)(x)\) also excludes \(x = 0\). So, when tackling function compositions, always remember to take their domains into account.

Function composition is a vital concept in algebra that involves applying one function to the results of another. Represented as \((f \circ g)(x) = f(g(x))\), it essentially means the output of function \(g(x)\) becomes the input of function \(f(x)\).

For the problem given, where \(f(x) = 2 - x\) and \(g(x) = \frac{1}{x^2}\), when you compose \(f\) and \(g\), you substitute the expression for \(g(x)\) into \(f(x)\). This results in \((f \circ g)(x) = 2 - \frac{1}{x^2}\).

But it's not just about substitution; understanding each function's role in the composition is also key. \(g(x)\) modifies the input by inverting and squaring it, and then \(f(x)\) further modifies the result by subtracting from 2. Such layers of operations demonstrate how complex functions can be built from simpler ones, which is the essence of composition.

Algebraic functions include expressions like polynomials, rational functions, and radicals, which can be constructed using algebraic operations such as addition, subtraction, multiplication, division, and extraction of roots. These functions form the backbone of many problems in mathematics.

In our exercise, both \(f(x) = 2 - x\) and \(g(x) = \frac{1}{x^2}\) are algebraic functions. \(f(x)\) is a polynomial of degree one, making it a simple linear function, while \(g(x)\) is a rational function, specifically a fraction with a polynomial in the denominator.

Understanding their properties individually aids in comprehending how they behave when combined in compositions. When a function like \(g(x) = \frac{1}{x^2}\) which involves division, is combined with \(f(x)\), the resulting function \((f \circ g)(x) = 2 - \frac{1}{x^2}\) still shares properties of both algebraic parents, possessing linearity from \(f\) and rational characteristics from \(g\). This inheritance of properties is a key feature of algebraic functions.