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April 12th, 2012, 06:35 PM   #1
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Number theory and finding the domain and range!

Hi I have three questions, two on number theory and the other on finding range and domain.

1.) Find the largest domain possible for the function f(x)-> (2)/(2x^2 -6x+5) and find the range corresponding to this domain.

I would like to know, is there any easy, shortcut method to do this? I have tried, it took a long time and my answer came out incorrect. What I did was to manually sub in numbers for x and then derive a conclusion from there. It is tedious and I am aware that I cant do this in an exam. How would you solve it?

2.) a,b and c are consecutive terms of both an AP and GP. Therefore a=b=c
I don't understand why a=b=c. Surely the value must increase or decrease from a to c? Unless they're talking about the difference, but I don't see that in the question.

3.) 1+((3/2)x)+((3/2)x)^2+((3/2)x)^3+...
I know that the common ratio is r=(3/2)x
The question asks, write down the range of values for x for the series to converge. The answer was -1<r<1
The rest of the steps I know how to continue, but I do not understand why for the series to converge, r has to be -1<r<1? I had expected -1<x<1
Also, what is the value of r happen if the series is to diverge? Someone told me r>0, is that right?

Thank you so much!
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April 12th, 2012, 07:08 PM   #2
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Re: Number theory and finding the domain and range!

1.) Equate the denominator to zero to find what values (if any) to exclude from the set of reals for the domain:





Thus, we find no real numbers to exclude. Thus the domain is:



Now to find the range, the function will have its maximal value when the denominator is at its minimum, which will occur at the axis of symmetry:



At this value of x the denominator is and thus

Since the function has a horizontal asymptote of , the range is then:



[attachment=0:44r3687v]range.jpg[/attachment:44r3687v]

2.) For a, b and c to be consecutive terms in an arithmetic progression, we require:



For a, b and c to be consecutive terms in a geometric progression, we require:



Thus, we have:





Hence:











3.) Check out this article on the condition for convergence:

http://en.wikipedia.org/wiki/Geometric_series
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April 12th, 2012, 08:41 PM   #3
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Re: Number theory and finding the domain and range!

Thank you so much!!
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