# quasi-inverse of a function

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

1. 1.

$g:Z\to X$ where $\operatorname{ran}(f)\subseteq Z\subseteq Y$, and

2. 2.

Note that $\operatorname{ran}(f)$ is the range of $f$.

Examples.

1. 1.

If $f$ is a real function given by $f(x)=x^{2}$. Then $g(x)=\sqrt{x}$ defined on $[0,\infty)$ and $h(x)=-\sqrt{x}$ also defined on $[0,\infty)$ are both quasi-inverses of $f$.

2. 2.

If $f(x)=1$ defined on $[0,1)$. Then $g(x)=\frac{1}{2}$ defined on $\mathbb{R}$ is a quasi-inverse of $f$. In fact, any $g(x)=a$ where $a\in[0,1)$ 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 $a\in[0,1)$, is a quasi-inverse of $f$.

Remarks.

## References

Title quasi-inverse of a function QuasiinverseOfAFunction 2013-03-22 16:22:14 2013-03-22 16:22:14 CWoo (3771) CWoo (3771) 11 CWoo (3771) Definition msc 03E20 quasi-inverse quasi-inverse function