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Partly undid revision 1097911133 by Vanadium-3065 (talk): some functions do have a close-form inverse; none of the given citations elaborates on approximability in general |
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===Left and right inverses===
[[Function composition]] on the left and on the right need not coincide. In general, the conditions
Left and right inverses are not necessarily the same. If {{mvar|g}} is a left inverse for {{mvar|f}}, then {{mvar|g}} may or may not be a right inverse for {{mvar|f}}; and if {{mvar|g}} is a right inverse for {{mvar|f}}, then {{mvar|g}} is not necessarily a left inverse for {{mvar|f}}. For example, let {{math|''f'': '''R''' → {{closed-open|0, ∞}}}} denote the squaring map, such that {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} for all {{mvar|x}} in {{math|'''R'''}}, and let {{math|{{mvar|g}}: {{closed-open|0, ∞}} → '''R'''}} denote the square root map, such that {{math|''g''(''x'') {{=}} }}{{radic|{{mvar|x}}}} for all {{math|''x'' ≥ 0}}. Then {{math|1=''f''(''g''(''x'')) = ''x''}} for all {{mvar|x}} in {{closed-open|0, ∞}}; that is, {{mvar|g}} is a right inverse to {{mvar|f}}. However, {{mvar|g}} is not a left inverse to {{mvar|f}}, since, e.g., {{math|1=''g''(''f''(−1)) = 1 ≠ −1}}.▼
# "There exists {{mvar|g}} such that {{math|''g''(''f''(''x'')){{=}}''x''}}" and
# "There exists {{mvar|g}} such that {{math|''f''(''g''(''x'')){{=}}''x''}}"
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====Left inverses====
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