Identity function in the context of Bijective function

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set.

A function is bijective if it is invertible; that is, a function ${\displaystyle f:X\to Y}$ is bijective if and only if there is a function ${\displaystyle g:Y\to X,}$ the inverse of f, such that each of the two ways for composing the two functions produces an identity function: ${\displaystyle g(f(x))=x}$ for each ${\displaystyle x}$ in ${\displaystyle X}$ and ${\displaystyle f(g(y))=y}$ for each ${\displaystyle y}$ in ${\displaystyle Y.}$

In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function. The elements of the isometry group are sometimes called motions of the space.

Every isometry group of a metric space is a subgroup of isometries. It represents in most cases a possible set of symmetries of objects/figures in the space, or functions defined on the space. See symmetry group.