According to wikipedia : "If the vertical line you drew intersects the graph more than once for any value of x then the graph is not the graph of a function." But I read on textbook this: "The point is that the graph of multi-valued function has the property of doubling back. In other words, this is what is implied when we say that a line parallel to the y axis intersects the graph in more than one point.... Notice that these points of tangency partition the curve into a union of single valued curves" I don't get it. Does anybody know better explain?
2026-05-06 04:15:25.1778040925
Vertical line test and multi-valued function
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There exists two different concepts of function here.
The wikipedia article uses the concept of a function where it takes in a number (or other mathematical object), or bunch of numbers, and returns a single number. Thus, each function will pass a vertical line test, and the wikipedia sentence holds for that concept.
A multi-valued function uses the concept where values of the domain get associated to values in a co-domain. There exists no limitation like with the wikipedia concept of a function. So, many values could get associated to '6' for some function.
The textbook talking about the union of single valued curves implies that a multi-valued function (on numbers, of course), could get talked about in terms of a union of single-valued functions, where single-valued refers to the number of outputs that a function can produce.