In calculus, a function is a rule that assigns a unique output to each input. The output is determined by the input, so we say that the output is a function of the input. For example, the area of a square is determined by its side length, so we say that the area is a function of its side length. Functions are used to describe relationships between two sets and are represented through tables, formulas, and graphs.
A formal definition of a function is that it is a mapping that associates the elements of one set, called the domain of the function, with the elements of another set, called the range of the function. For each value selected from the domain of the function, there exists exactly one corresponding element in the range of the function. The definition of the function tells us which element in the range corresponds to the element we picked from the domain.
Functions are essential in calculus and are used to describe physical relationships in the sciences. They are also used to model real-world phenomena, such as the motion of a bird or the growth of a plant.
