quasi-inverse of a function

Let f:XY be a function from sets X to Y. A quasi-inversePlanetmathPlanetmath g of f is a function g such that

  1. 1.

    g:ZX where ran(f)ZY, and

  2. 2.

    fgf=f, where denotes functionalPlanetmathPlanetmathPlanetmath composition operation.

Note that ran(f) is the range of f.


  1. 1.

    If f is a real function given by f(x)=x2. Then g(x)=x defined on [0,) and h(x)=-x also defined on [0,) are both quasi-inverses of f.

  2. 2.

    If f(x)=1 defined on [0,1). Then g(x)=12 defined on is a quasi-inverse of f. In fact, any g(x)=a where a[0,1) will do. Also, note that h(x)=x on [0,1) is also a quasi-inverse of f.

  3. 3.

    If f(x)=[x], the step function on the reals. Then by the previous example, g(x)=[x]+a, any a[0,1), is a quasi-inverse of f.


  • Every function has a quasi-inverse. This is just another form of the Axiom of ChoiceMathworldPlanetmath. In fact, if f:XY, then for every subset Z of Y such that ran(f)Z, there is a quasi-inverse g of f whose domain is Z.

  • However, a quasi-inverse of a function is in general not unique, as illustrated by the above examples. When it is unique, the function must be a bijection:

    If ran(f)Y, then there are at least two quasi-inverses, one with domain ran(f) and one with domain Y. So f is onto. To see that f is one-to-one, let g be the quasi-inverse of f. Now suppose f(x1)=f(x2)=z. Let g(z)=x3 and assume x3x1. Define h:YX by h(y)=g(y) if yz, and h(z)=x1. Then h is easily verified as a quasi-inverse of f that is different from g. This is a contradition. So x3=x1. Similarly, x3=x2 and therefore x1=x2.

  • Conversely, if f is a bijection, then the inversePlanetmathPlanetmathPlanetmathPlanetmath of f is a quasi-inverse of f. In fact, f has only one quasi-inverse.

  • The relationMathworldPlanetmath of being quasi-inverse is not symmetricPlanetmathPlanetmath. In other words, if g is a quasi-inverse of f, f need not be a quasi-inverse of g. In the second example above, h is a quasi-inverse of f, but not vice versa: h(0)=0, but hfh(0)=hf(0)=h(1)=1h(0).

  • Let g be a quasi-inverse of f, then the restrictionPlanetmathPlanetmathPlanetmath of g to ran(f) is one-to-one. If g and f are quasi-inverses of one another, and g strictly includes ran(f), then g is not one-to-one.

  • The set of real functions, with additionPlanetmathPlanetmath defined element-wise and multiplication defined as functional composition, is a ring. By remark 2, it is in fact a Von Neumann regular ringMathworldPlanetmath, as any quasi-inverse of a real function is also its pseudo-inverse as an element of the ring. Any space whose ring of continuous functions is Von Neumann regular is a P-space.


Title quasi-inverse of a function
Canonical name QuasiinverseOfAFunction
Date of creation 2013-03-22 16:22:14
Last modified on 2013-03-22 16:22:14
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 03E20
Synonym quasi-inverse
Defines quasi-inverse function