Equations VS. Functions - Algebra-1

1. Fundamental Concepts

Equations: Mathematical statements that express the equality between two expressions, with the core being "equality relationship". For example, \(2 + 3 = 5\) or those containing variables like \(x + y = 4\). Equations do not require a unique corresponding relationship between variables; one input can correspond to multiple outputs.
Functions: A special type of relationship where each input (independent variable) has a unique output (dependent variable) corresponding to it. It emphasizes "uniqueness" and can be regarded as an equation that meets specific conditions.

Essential Difference: Equations only reflect the equality relationship and do not restrict one input from corresponding to multiple outputs; while functions must satisfy the "single - value correspondence", that is, each independent variable can only correspond to one dependent variable.
Judgment Basis: To determine whether a relationship is a function, we need to check if there is a situation where "one input corresponds to multiple outputs". For example, in the set of points \((3, z)\) and \((3, 6)\), the input "3" corresponds to two outputs, so it is not a function.

Equations that are not functions: Such as \(x^2 + y^2 = 4\) (the equation of a circle). When \(x = 0\), \(y = 2\) or \(y=-2\), one input corresponds to two outputs, so it is not a function.
Examples of functions: Such as \(y = 2x\). For any input x, there is a unique y corresponding to it, which conforms to the definition of a function.

Method to judge a function: It can be combined with the "Vertical Line Test". That is, if a vertical line perpendicular to the x - axis intersects the graph at exactly one point, the relationship represented by the graph is a function; if there are multiple intersection points, it is not a function.
Distinguishing between equations and functions: Analyze the relationship between variables. If there is "one input corresponding to multiple outputs", it is only an equation rather than a function; if it satisfies "single - value correspondence", it is both an equation and a function.