function 

x ↦ f (x) 
examples by domain and codomain 

x  → 
b,  b 
→  x, 
b^{n}  →  b 
x  → 
z,  z 
→  x 
x  → 
r,  r 
→  x, 
r^{n}  →  x 
x  → 
c,  c 
→  x, 
c^{n}  →  x 

classes/properties 

constant · identity · linear · polynomial · rational · algebraic · analytic · smooth · continuous · measurable · injective · bijective 
constructions 

restriction · composition · λ · inverse 
generalizations 

partial · multivalued · implicit 

a bijective function,
f:
x →
y, where set x is {1, 2, 3, 4} and set y is {a, b, c, d}. for example,
f(1) = d.
in mathematics, a bijection, bijective function, onetoone correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. there are no unpaired elements. in mathematical terms, a bijective function f: x → y is a onetoone (injective) and onto (surjective) mapping of a set x to a set y.^{[1]}^{[2]} the term onetoone correspondence must not be confused with onetoone function (a.k.a. injective function) (see figures).
an injective nonbijection)
an injective surjective function (bijection)
a noninjective surjective function (bijection)
a noninjective nonsurjective function (also not a bijection)
a bijection from the set x to the set y has an inverse function from y to x. if x and y are finite sets, then the existence of a bijection means they have the same number of elements. for infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets.
a bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms a symmetry group.
bijective functions are essential to many areas of mathematics including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group, and projective map.