Answer:
The inverse of a non-function mapping is not necessarily a function.
For example, the inverse of the non-function mapping
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
. 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
, at most one of the distinct pairs includes
as the value of first entry.
For example, the mapping
is a function. However, the mapping
isn't a function since more than one of the distinct pairs in this mapping include
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
is the mapping
.
Consider mapping
. This mapping isn't a function since the input value
is the first entry of more than one of the pairs.
Invert
as follows:
becomes
.
becomes
.
becomes
.
becomes
.
In other words, the inverse of the mapping
would be
, which is the same as the original mapping. (Mappings are sets. There is no order between entries within a mapping.)
Thus,
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
mappings, including circles) are also non-function mappings.