Algebra.Com's Answer #849677 by ikleyn(52915)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
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document.write( "The domain of the function r(x) = (x^2 + 1)/(1 - x)^2 is (-infty,1) U (1,infty). What is the range?
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document.write( " I will provide another solution, which uses Algebra only\r
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document.write( " and does not use Calculus.\r
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document.write( "Real number \"t\" belongs to the range of this function if and only if \r\n" );
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document.write( "  = t for some real x. (1)\r\n" );
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document.write( "Rewrite (1) in an equivalent form\r\n" );
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document.write( "  = t,\r\n" );
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document.write( "  =  ,\r\n" );
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document.write( " x^2 + 1 = tx^2 - 2tx + t,\r\n" );
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document.write( " (1-t)x^2 + 2tx + (1-t) = 0.\r\n" );
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document.write( "This last quadratic equation has a real solution IF and ONLY IF the discriminant is non-negative\r\n" );
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document.write( " d = b^2 - 4ac >= 0,\r\n" );
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document.write( "or\r\n" );
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document.write( " (2t)^2 - 4*(1-t)*(1-t) >= 0,\r\n" );
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document.write( " 4t^2 - 4*(1-t)^2 >= 0,\r\n" );
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document.write( " t^2 - 1 + 2t - t^2 >=0,\r\n" );
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document.write( " 2t >= 1,\r\n" );
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document.write( " t >= 1/2.\r\n" );
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document.write( "At this point, the solution is complete.\r\n" );
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document.write( "ANSWER. The domain is the set of all real numbers greater than or equal to 1/2.\r\n" );
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document.write( "Solved.\r
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document.write( "The solution and the answer in the post by Edwin are incorrect.\r
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