Its domain is the set of all real numbers different from formula_204 and its image is the set of all real numbers different from formula_205 If one extends the real line to the projectively extended real line by including , one may extend to a bijection from the extended real line to itself by setting formula_206 and formula_207. A multivariate function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time and its speed. More formally, a function of variables is a function whose domain is a set of -tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. More generally, every mathematical operation is defined as a multivariate function. The Cartesian product formula_208 of sets formula_209 is the set of all -tuples formula_210 such that formula_211 for every with formula_212. Therefore, a function of variables is a function where the domain has the form When using function notation, one usually omits the parentheses surrounding tuples, writing formula_215 instead of formula_216  In the case where all the formula_217 are equal to the set formula_218 of real numbers, one has a function of several real variables. If the formula_217 are equal to the set formula_220 of complex numbers, one has a function of several complex variables. It is common to also consider functions whose codomain is a product of sets. For example, Euclidean division maps every pair of integers with to a pair of integers called the "quotient" and the "remainder": The codomain may also be a vector space. In this case, one talks of a vector-valued function. If the domain is contained in a Euclidean space, or more generally a manifold, a vector-valued function is often called a vector field. The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus (see History of the function concept). At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth. But the definition was soon extended to functions of several variables and to functions of a complex variable.