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Is it true that if curves are self-intersecting, then function graph would be non injective ?

Like suppose a differential curve $\alpha(t) = (x,y) = (t^2, t^3) $ where $y=f(x) ,$ its function graph comes out as $ y=x^{3/2} $ and they self-intersect and so are non injective?

Let us take another functions like $\alpha(t) = (3t/1+t^3, 3t^2+1+t^3) $ never self-intersect and they are injectives. So my question is:

Do intersections solly define a function's injectivity or non-injectivity ? What are other definitive requirements? Thanks.

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  • $\begingroup$ Welcome to MSE. Questions are usually received better if you give a bit more background including what you have tried and where you are stuck. This should be added to the question itself. $\endgroup$ Jul 18, 2018 at 12:09
  • $\begingroup$ Please Check out the question once again i put some explanation here. I'm trying Differential Geometry and it's about curves. $\endgroup$ Jul 18, 2018 at 12:19

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You may use horizontal lines to verify that a function is injective.

If every horizontal line intersect the graph at most once the function is injective

Otherwise it is not injective

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  • $\begingroup$ Can you tell me suppose functions alpha(t) = t^3-4t, t^2 -4. Would it be injective ? $\endgroup$ Jul 18, 2018 at 12:35
  • $\begingroup$ Where t1,2 = +-2 $\endgroup$ Jul 18, 2018 at 12:43
  • $\begingroup$ $alpha(t) = t^3-4t, t^2 -4 $ are not injective. $\alpha (-2)=\alpha (2)$ $\endgroup$ Jul 18, 2018 at 13:02
  • $\begingroup$ Thanks ! It was helpful. My doubts are cleared now. $\endgroup$ Jul 18, 2018 at 15:13

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