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January 13th, 2019, 08:24 AM   #1
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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.
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January 13th, 2019, 08:42 AM   #2
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Quote:
Originally Posted by idontknow View Post
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.
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Last edited by skipjack; January 13th, 2019 at 09:58 PM.
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