2}. (In fact, the pre-image of this function for every y, −2 ≤ y ≤ 2 has more than one element.) The function defined by g(x) = x2 is not surjective, since there is no real number x such that x2 = −1. However, the function defined by (with the restricted codomain) is surjective, since for every y in the nonnegative real codomain Y, there is at least one x in the real domain X such that x2 = y. The natural logarithm function is a surjective and even bijective (mapping from the set of positive real numbers to the set of all real numbers). Its inverse, the exponential function, if defined with the set of real numbers as the domain, is not surjective (as its range is the set of positive real numbers). The matrix exponential is not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as a map from the space of all n×n matrices to the general linear group of degree n (that is, the group of all n×n invertible matrices). Under this definition, the matrix exponential is surjective for complex matrices, although still not surjective for real matrices. The projection from a cartesian product to one of its factors is surjective, unless the other factor is empty. In a 3D video game, vectors are projected onto a 2D flat screen by means of a surjective function. Properties A function is bijective if and only if it is both surjective and injective. If (as is often done) a function is identified with its graph, then surjectivity is not a property of the function itself, but rather a property of the mapping. This is, the function together with its codomain. Unlike injectivity, surjectivity cannot be read off of the graph of the function alone. Surjections as right invertible functions The function is said to be a right inverse of the function if f(g(y)) = y for every y in Y (g can be undone by f). In other words, g is a right inverse of f if the composition of g and f in that order is the identity function on the domain Y of g. The function g need not be a complete inverse of f because the composition in the other order, , may not be the identity function on the domain X of