This becomes even more evident when trying to solve complex equations or equations with more than one variable.

For example, when determining the solution to an equation, such as 3x + 4y = 8, the two resulting numbers—called a set of ordered pairs of numbers—is called a relation. In turn, a function then becomes a relation in which each first element, such as x, is matched exactly with a second element, such as y. In other words, a function can take on a definite value (or values) when certain values are assigned to other quantities or independent variables of the function. (For more information about functions, see “Foundations of Mathematics” and “Algebra.”)

A one-to-one function is one in which each input (number that replaces the variable) has exactly one output (result of the equation). In such cases, the function needs to pass the “horizontal line test,” showing that a horizontal line intersects the graph once and only once. For example, for the equation f(x) = x², if one restricts the answer to x ≥ 0, the result is a one-to-one function; but the equation with no constraints is not a one-to-one function, as the output value of 4 has the two input values of 2 and -2.