Question

Could the inverse of a non-function be a function? Explain or give an example.

Answer 1

The inverse of a non-function mapping is not necessarily a function.

For example, the inverse of the non-function mapping
`\lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\!` is the same as itself (and thus isn't a function, either.)

Explanation:

A mapping is a set of pairs of the form
`(a,\, b)`. 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
`x`, at most one of the distinct pairs includes
`x\!` as the value of first entry.

For example, the mapping
`\lbrace (0,\, 0),\, (1,\, 0) \rbrace` is a function. However, the mapping
`\lbrace (0,\, 0),\, (1,\, 0),\, (1,\, 1) \rbrace` isn't a function since more than one of the distinct pairs in this mapping include
`1` 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
`\lbrace (a_(1),\, b_(1)),\, (a_(2),\, b_(2))\rbrace` is the mapping
`\lbrace (b_(1),\, a_(1)),\, (b_(2),\, a_(2))\rbrace`.

Consider mapping
`\lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\!`. This mapping isn't a function since the input value
`0` is the first entry of more than one of the pairs.

Invert
`\lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\!` as follows:

• 
  `(0,\, 0)` becomes
  `(0,\, 0)`.
• 
  `(0,\, 1)` becomes
  `(1,\, 0)`.
• 
  `(1,\, 0)` becomes
  `(0,\, 1)`.
• 
  `(1,\, 1)` becomes
  `(1,\, 1)`.

In other words, the inverse of the mapping
`\lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\!` would be
`\lbrace (0,\, 0),\, (1,\, 0),\, (0,\, 1),\, (1,\, 1) \rbrace\!`, which is the same as the original mapping. (Mappings are sets. There is no order between entries within a mapping.)

Thus,
`\lbrace (0,\, 0),\, (0,\, 1),\, (1,\, 0),\, (1,\, 1) \rbrace\!` 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
`\mathbb{R} \to \mathbb{R}` mappings, including circles) are also non-function mappings.