Functions and vertical line proof

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

(a) What happens with this?:

(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 ...

• 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. – Git Gud Aug 9 '13 at 22:19
• ... but of you view $x$ as a function of $y$, go ahead. – Hagen von Eitzen Aug 9 '13 at 22:20
• 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! – Alex Wertheim Aug 9 '13 at 23:00

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.