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 Can someone give me a hint to how I should approach this.  I think the best way is guess and check and see the difference but there must be a better way.  Problem: consider the function f(x)= sqrt(x+1/x-1) and g(x)=sqrt(x+1)/sqrt(x-1). Explain why f and g arent the same?

Guest Jul 18, 2017
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 #1
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At first glance, these two look the same -- after all, one could simplify  \(\sqrt{\frac{x+1}{x-1}}\)into \(\frac{\sqrt{x+1}}{\sqrt{x-1}}\).

However, if we try to graph these using an x-y table:

x-y table for first equation
x y
-2 \(\sqrt{\frac{-1}{-3}} = \sqrt{\frac{1}{3}} = \frac{\sqrt{3}}{3}\)
-1 \(\sqrt{\frac{-1+1}{-1-1}}=\sqrt{\frac{0}{-2}}=0\)
0 (undefined)
1 (undefined, asymptote)
2 \(\sqrt{\frac{3}{1}}=\sqrt{3}\)
x-y table for second equation
x y
-2 undefined
-1 undefined
0 undefined
1 undefined, asymptote
2 \(\sqrt{\frac{3}{1}}=\sqrt{3}\)
3 ...

 

As you can see, \(\sqrt{\frac{x+1}{x-1}}\)allows for negative values, but \(\frac{\sqrt{x+1}}{\sqrt{x-1}}\)doesn't.

My advice: when in doubt, plug in a few numbers of graph it. 

Guest Jul 18, 2017
 #2
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This graph should help (note: for x>=1 the two are the same):

 

Alan  Jul 18, 2017
edited by Alan  Jul 18, 2017

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