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Hi, can someone help me with these two problems? (Please also explain how you got the answers)

 

1. Let \(f(x) : (-\infty,0) \cup (0,\infty) \to \mathbb{R}\) be defined by f(x) = x - 1/x. Show that f(x) has no inverse function.

2. Let \(g(x) : (0,\infty) \to \mathbb{R}\) be defined by g(x) = x - 1/x. Show that g(x) has an inverse function.

 

Thank you so much!

 Dec 22, 2020
edited by Caffeine  Dec 22, 2020
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If you graph y = x - 1/x, then it clearly fails the Horizontal Line Test over all real numbers, so f is not invertible.  But if you restrict to positive real numbers, then it passes the Horizontal LIne Test, so g is invertible.

 Dec 22, 2020

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