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