Is this called a partial inverse function?

Question

Good day everyone. Here goes my question:

Let's take as example the function $f(x) = 2x$ defined over naturals. The function is injective and has an inverse $f^{-1}(y) = \frac{y}{2}$ which is only defined for even numbers, that is, the image of $f$ is a subset of $Y$ and the inverse only works for the elements of this subset.

What is the name of such inverse function? Partial inverse function?

Thanks a lot

Esteve

Answer 1

I'm not aware of any name for this function. You could simply say that the image of $f$ is a (proper) subset of the domain of $f$. It's certainly not a partial function: a partial function is a relation between two sets $A$ and $B$ such that each member of $A$ is associated to at most one element of $B$; this is in contrast to the definition of a function, where each member of $A$ is associated to exactly one element of $B$. For instance, $f:\mathbb R\to\mathbb R,f(x)=\sqrt{x}$ is a partial function, as $\sqrt{x}$ is only defined when $x\ge0$.

Answer 2

That depends on how the function is defined.

A function is a special kind of binary relations.

Wikipedia - Function (mathematics):
"In mathematics, a function from a set $X$ to a set $Y$ assigns to each element of $X$ exactly one element of $Y$. The set $X$ is called the domain of the function and the set $Y$ is called the codomain of the function.
...
Two functions $f$ and $g$ are equal if their domain and codomain sets are the same and their output values agree on the whole domain.
...
The [...] and codomain are not always explicitly given when a function is defined, and, without some (possibly difficult) computation, one might only know that the domain is contained in a larger set."

We have to distinguish the terms image and codomain. The image is a (proper or improper) subset of the codomain. If the function is bijective (surjective and injective), the inverse function exists. If the inverse function exists, the image is the domain of the inverse function.

You need the terms surjective function, partial function, restriction, partial inverse and codomain-restriction.
A function is trivially surjective when it is restricted to its image.

In the definition of a function, we can specify a codomain that is the image (a) or a codomain that is a proper superset of the image (b):

a)
$f_a\colon\mathbb{N}\to\{s\mid (s\in\mathbb{N}) \land (2|s)\}, x\mapsto 2x$
$f_a^{-1}\colon\{t\mid (t\in\mathbb{N}) \land (2|t)\}\to\mathbb{N}, y\mapsto\frac{y}{2}$
The function $f_a$ is injective and surjective, therefore the inverse relation $f_a^{-1}$ is a function, the inverse function.

b)
$f_b\colon\mathbb{N}\to\mathbb{R}_{\ge 0}, x\mapsto 2x$
$f_b^{-1}\colon\mathbb{R}_{\ge 0}\to\mathbb{N}, y\mapsto\frac{y}{2}$
The function $f_b$ is not surjective, therefore the inverse relation $f_b^{-1}$ is not a function, it is a partial function.

The function $f_a$ is a subfunction of the function $f_b$, a codomain-restriction of $f_b$.
The function $f_a^{-1}$ is a subrelation of the relation $f_b^{-1}$, a restriction of $f_b^{-1}$.
The function $f_a^{-1}$ is a partial inverse function of the function $f_b$.

see e.g. the tables in Wikipedia German: Relation (Mathematik) - Eigenschaften zweistelliger Relationen