document.write( "Question 1189172: (a) Let f : (-\infty,0) \cup (0,\infty) \to \mathbb{R} be defined by
\n" ); document.write( "f(x) = x - \frac{1}{x}.Show that f has no inverse function.\r
\n" ); document.write( "\n" ); document.write( "(b) Let g : (0,\infty) \to \mathbb{R} be defined by
\n" ); document.write( "g(x) = x - \frac{1}{x}. Show that g has an inverse function.\r
\n" ); document.write( "\n" ); document.write( "How can you show this without graphing.
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Algebra.Com's Answer #820476 by ikleyn(52782)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We are given the function  f(x) = x - \"1%2Fx\".\r\n" );
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document.write( "In part (a), the domain of this function is  (-oo,0) U (0,oo).\r\n" );
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document.write( "Notice that there is an identity  f(x) = \"f%28-1%2Fx%29\",  valid for all x from the domain  (check it on your own).\r\n" );
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document.write( "It means that each image of f(x) has two pre-images: x and \"-1%2Fx\".\r\n" );
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document.write( "One of these pre-images is positive number; the other pre-image is negative number.\r\n" );
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document.write( "So, the function f(x), defined on the whole domain, is not one-to-one function; \r\n" );
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document.write( "THEREFORE, it has no an inverse function.\r\n" );
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document.write( "In part (b), the domain is restricted: now it is the set of all positive numbers.\r\n" );
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document.write( "On this set, the function f(x) = x - \"1%2Fx\" is monotonically increasing  \r\n" );
document.write( "(check it on your own, for example, taking the derivative).\r\n" );
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document.write( "Hence, it is one-to-one map.\r\n" );
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document.write( "Therefore, on the restricted domain, this function has an inverse function.\r\n" );
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