In order for an equation to be a function, it has to pass the vertical line test.(adsbygoogle = window.adsbygoogle || []).push({});

A circle is not a function because it does not pass the vertical line test.

A curve containing a loop does not pass the vertical line test and to me that means it is not a function.

However, if I am given two parametric equations, x(t) and y(t) that both seem to be functions (passing the vertical line test), and am provided a picture of the graph of the parametric equations, if the curve of the parametric equations contains a loop, I would assume right away that the combined equation, y(x), can not be a function because the graph has a loop.

So if I have x(t) = t^{5} + 4t^{3} and y(t) = t^{2}

And because x(t) is not solvable, I can't sub the t(x) into y(t), but I can sub t(y) into x(t):

[itex]x(y) = ((+/-)y^{5/2}- 4(+/-)y^{3/2}[/itex]

[itex]x= y^{5/2}- 4y^{3/2}[/itex] or [itex]x= -y^{5/2}+ 4y^{3/2}[/itex]

My question is here:

Is there anything specific about this function or a function of similar form that enables me to see that A) it is not a function B) It creates a loop when graphed (implying there is at least some point at which the curve intersects itself)?

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# A curve that intersects itself at some point w/o trig functions

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