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Non-functions and the vertical line test

  1. Aug 1, 2008 #1
    Why does the vertical line test work? If the line hits the graph more than once, it is said to not be a function. But why?
  2. jcsd
  3. Aug 1, 2008 #2


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    Hi DecayProduct ! :smile:

    Because a map is a function from one space to another …

    so, for any point x in the first space, there must be an f(x) in the second space.

    There can't be more than one f(x) for the same x.

    And if the (vertical) line hit the graph more than once, then that would make two f(x)'s for one x! :smile:
  4. Aug 1, 2008 #3


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    The definition of a function is "a set of ordered pairs (x, f(x)) so that now two pairs have the same first member" or, equivalently, a mapping x-> f(x) so that the same x does not give two different values for f(x).

    The graph of a relation consists of the points (x,f(x)). A vertical line corresponds to a specific value of x. If a vertical line crosses the graph at two points, that means we have two different values of y associated with the same value of x: f(x)= y1 and f(y)= y2 for different y1 and y2. That violates the definition of "function".
  5. Aug 1, 2008 #4
    Thanks to both of you! That makes sense. I theorized that, but not so eloquently, and of course, wasn't sure if that was even close to why.
  6. Aug 2, 2008 #5
    A curve in the xy plane is the graph of some function ƒ IF no vertical line intersects the curve more than once.

    You're right, but don't confuse the "vertical line test" with the "horizontal line test", this is obviously one of the mistakes I made when I started calculus.

    The horizontal line test tells you wether a function is one to one and onto or not (ie. a horizontal line can't intersect a bijective function "1:1" more than once)

    Be careful with that! :tongue:
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