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

#1**+1 **

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.

Guest Jul 18, 2017