How do you draw graphs of fractions? And how do you know when to put "U" in Domain?

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username970964
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#1
Report Thread starter 7 years ago
#1
So say the fraction was x+9/x^2-4. How do you work out how to draw the graph. I worked out the domain to be {xIx≠-2 or 2}. But if you were to set out the domain in a different way, to put Union, "U" in it. How do you know where to put it in?

The union of two sets is everything in both sets. So to join them together you put a U in. But how do you know when the two sets are separated?
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Ktulu666
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#2
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Not quite sure what you mean about the domain (as far as I can see, you'd need a union of three sets, like {x: x<-2}U{x: -2<x<2}U{x: x>2} though the motivation for this escapes me. Basically, in order to draw a graph, write down everything you know about the graph. Any vertical/horizontal asymptotes? What about oblique ones (what is the behaviour of the function when x -> +/- infinity)? Intersection points with the axes, stationary points, just deduce things about the function from its equation and mark them on the axes. Eventually you'll collate enough information to be able to interpolate and draw the whole graph.

EDIT: You know two sets are separated (disjoint) if AnB = {}, the empty set. For example with my sets, {x: x<-2}n{x: -2<x<2} = {}, since if x is in both sets, it must be simultaneously greater than and less than -2, which contradicts the order axioms of the real numbers.
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Smaug123
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#3
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(Original post by Airess3)
So say the fraction was x+9/x^2-4. How do you work out how to draw the graph. I worked out the domain to be {xIx≠-2 or 2}. But if you were to set out the domain in a different way, to put Union, "U" in it. How do you know where to put it in?

The union of two sets is everything in both sets. So to join them together you put a U in. But how do you know when the two sets are separated?
I think you should find out what a union actually is, from a teacher or something. It sounds like you've just learnt that the union should be used in this case, without knowing what on earth a union is.
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Hasufel
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#4
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#4
this really depends on the function, and personal preference for explaining domain.

For instance, in the function  \displaystyle \frac{1}{x} the domain is better described as:

x\in \mathbb{R}:x\neq 0

for \displaystyle 1+ \frac{1}{ \sin(x)}

x\in \mathbb{R}:x\neq n \pi , n\in \mathbb{Z}

for rational functions where the denominator is a reducible (over the reals, lets say for ease) polynomial, then unions are more common the more points the function is undefined at

If you wish to sketch rational functions, then, assuming both numerator and denominator are polynomials, if you need to divide out if the degree of numerator > degree of denominator, then examine the behaviour of the function as x approaches +/- infinity, then as it approaches any point where it is undefined (where the denominator =0) from left and right
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username970964
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#5
Report Thread starter 7 years ago
#5
Thanks for the help, I know how to do it now.
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