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?
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.
