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 Calculus Calculus Math Forum

 January 13th, 2019, 08:24 AM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 636 Thanks: 91 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 09:56 PM. January 13th, 2019, 08:42 AM   #2
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Quote:
 Originally Posted by idontknow 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.
A closed figure is not the graph of a function.

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 09:58 PM. Tags function, graph Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post hy2000 Algebra 2 August 14th, 2018 12:02 AM hy2000 Algebra 7 August 13th, 2018 11:45 PM hlecok Algebra 1 September 5th, 2012 09:05 AM ChristinaScience Calculus 7 October 9th, 2011 11:35 AM mikeportnoy Algebra 3 March 10th, 2009 06:35 AM

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