January 13th, 2019, 09:24 AM  #1 
Senior Member Joined: Dec 2015 From: Earth Posts: 316 Thanks: 42  Graph of a function
Let $\displaystyle y=f(x)$ How can I know whether the graph of f(x) is a closed curve or not? Example: the circle is a closed curve, $\displaystyle y^2 +x^2 =R^2$ Without using the graph. Last edited by skipjack; January 13th, 2019 at 10:56 PM. 
January 13th, 2019, 09:42 AM  #2  
Senior Member Joined: May 2016 From: USA Posts: 1,245 Thanks: 515  Quote:
Therefore if you solve an equation of the form f(x) = g(y) for y in terms of x and find that that y exists only if the domain of x is bounded, that, except where the derivative is undefined, you get multiple values of y for a single value of x, and that y has a single value at the end points of x's domain, you are dealing with a closed figure. $\displaystyle x^2 + y^2 = r^2 \implies y = \pm \sqrt{r^2  x^2} \implies \frac{dy}{dx} = \pm \frac{x}{\sqrt{r^2  x^2}}.$ See that x is bounded by  r and r. At those boundary points, y has a single value, but at every other point in the domain, y has more than 1 possible value. At the end points, where y has a single value, the derivative is undefined. Last edited by skipjack; January 13th, 2019 at 10:58 PM.  

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