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  • https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/06%3A_Relations_and_Functions/6.06%3A_Special_Functions
    There are a great many functions that fail the horizontal line test which we nevertheless seem to have inverse functions for. For example, x^2 fails HLT but the square root of x is a pretty reasonabl...There are a great many functions that fail the horizontal line test which we nevertheless seem to have inverse functions for. For example, x^2 fails HLT but the square root of x is a pretty reasonable inverse for it – one just needs to be careful about the “plus or minus” issue. This apparent contradiction can be resolved using the notion of restriction.

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