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

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

 Nov 2, 2020
 #1
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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.

 

 Nov 2, 2020

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