Answer:
The inverse of a non-function mapping is not necessarily a function.
For example, the inverse of the non-function mapping [tex]\lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\![/tex] is the same as itself (and thus isn't a function, either.)
Step-by-step explanation:
A mapping is a set of pairs of the form [tex](a,\, b)[/tex]. The first entry of each pair is the value of the input. The second entry of the pair would be the value of the output.
A mapping is a function if and only if for each possible input value [tex]x[/tex], at most one of the distinct pairs includes [tex]x\![/tex] as the value of first entry.
For example, the mapping [tex]\lbrace (0,\, 0),\, (1,\, 0) \rbrace[/tex] is a function. However, the mapping [tex]\lbrace (0,\, 0),\, (1,\, 0),\, (1,\, 1) \rbrace[/tex] isn't a function since more than one of the distinct pairs in this mapping include [tex]1[/tex] as the value of the first entry.
The inverse of a mapping is obtained by interchanging the two entries of each of the pairs. For example, the inverse of the mapping [tex]\lbrace (a_{1},\, b_{1}),\, (a_{2},\, b_{2})\rbrace[/tex] is the mapping [tex]\lbrace (b_{1},\, a_{1}),\, (b_{2},\, a_{2})\rbrace[/tex].
Consider mapping [tex]\lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\![/tex]. This mapping isn't a function since the input value [tex]0[/tex] is the first entry of more than one of the pairs.
Invert [tex]\lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\![/tex] as follows:
In other words, the inverse of the mapping [tex]\lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\![/tex] would be [tex]\lbrace (0,\, 0),\, (1,\, 0),\, (0,\, 1),\, (1,\, 1) \rbrace\![/tex], which is the same as the original mapping. (Mappings are sets. There is no order between entries within a mapping.)
Thus, [tex]\lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\![/tex] is an example of a non-function mapping that is still not a function.
More generally, the inverse of non-trivial ellipses (a class of continuous non-function [tex]\mathbb{R} \to \mathbb{R}[/tex] mappings, including circles) are also non-function mappings.
