Question

Hi everyone! :o) Can someone look at the graph I have posted at Wolfram and tell me what the circle portions are? Not sure I have ever seen a graph like that with little circles on it. Thanks!
http://www.wolframalpha.com/input/?i=y^3-3y^2%3D1%2Fx%2Bx^3-2

Answer 1

@ganeshie8

Answer 2

I guess the real question I need to answer is whether or not this function is defined from -infinity to infinity or do those little circles mess things up somehow?

Answer 3

for it to be a function, it has to pass the vertical line test,

any vertical line ( | ) on the plane, can only cross the line of a function at one place;
if it crosses at multiple places, it cannot be a functioin

Answer 4

because a function at some input x, should only have one output value (y)

Answer 5

so what is the interval of definition then for this equation?

Answer 6

is it just not defined where a vertical line goes through those portions with the circles then?

Answer 7

yes, the relation has one defined value at the places where the vertical line test holds (like a function), and the places where the vertical line test fails, cannot be in an interval where the relation is well defined because there are a few possible values

Answer 8

your region of definition should be four separate intervals,

Answer 9

so if that equation was a solution to a differential equation, i could say this then>>> (-infinty, to the left of the first circle) , (to the right of the first circle, to the left of the second circle) and (to the right of the second circle, +infinty)

would that be correct logic but very ugly interval notation? :o)

Answer 10

you mean 3 separate intervals right? not 4?

Answer 11

there is another point that fails the vertical line test

Answer 12

nevermind, i forgot the asymptote! :o)
4 intervals!
thanks Unkle! here's your cookie!

Answer 13

i expect the region will not look nice in interval notation at all

Answer 14

that's right

Answer 15

thanks!

Answer 16

thats messed up, why is that happening?

Answer 17

because ` ^2 ` isn't one-to-one