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In mathematics, a function is a relation between a set of inputs and a set of permissible outputs. Each input is related to exactly one output. Think of it as a machine that takes some ingredient and spits out a product, and the same ingredient cannot result in two different products. In formal terms, for a relation to be considered a function, every element in the domain (all possible inputs) must be associated with a unique element in the range (all possible outputs).

This uniqueness is what distinguishes a function from a non-function. When working with an equation to determine if it defines a function, you must ensure that for every value of the independent variable (commonly represented as 'x'), there is at most one corresponding value of the dependent variable (often 'y').

For example, in the given equation from the exercise, the presence of both a positive and a negative result when solving for 'y' indicates that a single 'x' value is associated with more than one 'y' value. Hence, this violates the rule of one input to one output, thus indicating that the relation is not a function.

Solving equations is a fundamental skill in algebra that involves finding the value(s) of the variable(s) that make the equation true. It's like decoding a puzzle where you need to find the missing piece. The process usually involves manipulating the equation to isolate the variable on one side and the numbers on the other. There's a variety of tools in the mathematician's toolkit for solving equations, including addition, subtraction, multiplication, division, factoring, and taking roots.

For the quadratic equation in the exercise, the process begins by isolating one of the variables. After moving terms from one side to the other and eventually isolating the variable, we reach a point where we must extract the variable from inside a square—this is where square roots come into play.

Understanding the properties of equality is crucial when solving equations, especially knowing that what you do to one side of the equation, you must also do to the other to maintain balance. This is how the mystery is unwrapped and the values of the variables are revealed, provided the steps are logically and systematically applied.

A square root of a number is a value that, when multiplied by itself, gives the original number. Symbolically, the square root of a number 'a' is written as \(\sqrt{a}\), and if \(\sqrt{a} = b\), then \(b^2 = a\). Importantly, every positive number has two square roots: a positive and a negative root, because both \(+b\) and \(−b\) squared will result in the original number.

For instance, the number 25 has two square roots: \(+5\) and \(−5\), because both \(5^2\) and \(−5^2\) equal 25. This concept is pivotal when interpreting quadratic equations where two potential 'y' values emerge for each 'x' value, leading to a situation where a single input might suggest two outputs, which, as discussed previously, violates the definition of a function.

Understanding that taking the square root of an expression such as \(25 - x^2\) yields both \(+\sqrt {25 - x^2}\) and \(−\sqrt {25 - x^2}\), indicates the importance of considering the full scope of the square root's consequences when engaging with quadratic expressions and analyzing their roles in defining functions.