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A curve is a function if satisfy the vertical line test.

(a) What happens with this?:

enter image description here

(b) do not exist a function for graphing this curve? Really?

P.S. I have a TI graph calculator and I once make some graphic with it and I am almost sure that I made this curve. I really don`t recall how I doit ...

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    $\begingroup$ It is possible to draw such curve on the TI calculator, but not by inserting an expression in the $Y=$ menu, the reason is exactly what you stated in your question: such a curve isn't a graph. $\endgroup$
    – Git Gud
    Aug 9, 2013 at 22:19
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    $\begingroup$ ... but of you view $x$ as a function of $y$, go ahead. $\endgroup$
    – Hagen von Eitzen
    Aug 9, 2013 at 22:20
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    $\begingroup$ It is even possible to draw this curve on the TI calculator. Simply enter $y= x^{2}$ in the usual place, and then turn your calculator 90 degrees clockwise! $\endgroup$
    – Alex Wertheim
    Aug 9, 2013 at 23:00

2 Answers 2

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More accurately: A curve in the $xy$-plane is a function of the variable $x$ if and only if it satisfies the vertical line test. The graphed parabola is not a function of $x,$ because it fails the vertical line test. As a side note, it is still a function, but it is a function of $y$ (because it passes the horizontal line test). It is still possible to graph such a curve on a graphing calculator, specifically by graphing the upper and lower branches as two separate functions on the same graph.

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A graph of a function by this definition can only have one output for each input. If you were to express this in terms of x, it would look something like "plus or minus the square root of the quantity x minus k": y = f(x) = +-sqrt(x - h) The "plus or minus" gives you two outputs.

If you express the function in terms of y (ie "x = f(y)"), it will pass the horizontal line test.

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