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# Help

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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?

Jul 18, 2017

#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 -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 -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.

Jul 18, 2017
#2
+27374
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This graph should help (note: for x>=1 the two are the same):

Jul 18, 2017
edited by Alan  Jul 18, 2017