Algebra.Com's Answer #846167 by ikleyn(52798)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Find the range of f(x) = (x^2 + 2)/(x + 4) algebraically.
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document.write( "Let real number \"t\" belongs to the range. It means that\r\n" );
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document.write( "  = t (1)\r\n" );
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document.write( "for some value of x. Step by step, reduce equation (1) to standard form quadratic equation\r\n" );
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document.write( "  =  ,\r\n" );
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document.write( "  = 0. (2)\r\n" );
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document.write( "Equation (2) is a standard form quadratic equation  = 0 with coefficients\r\n" );
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document.write( " a = 1, b = -t, c = 2-4t.\r\n" );
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document.write( "The condition that it has a real solution for  is this inequality for the discriminant\r\n" );
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document.write( "  >= 0, or  >= 0, or t^2 + 16t -8 >= 0.\r\n" );
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document.write( "The roots of this quadratic equation for t are\r\n" );
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document.write( "  =  =  = -16.4853 (rounded) and  =  =  = 0.4853 (rounded).\r\n" );
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document.write( "The discriminant is non-negative if and only if value of \"t\" is out of the roots' interval \r\n" );
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document.write( " t <= = -16.4853 or t >= =0.48528.\r\n" );
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document.write( "Therefore, equation (1) has a real solution if and only if  < t <= or <= t <  .\r\n" );
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document.write( "Thus the range of the function f(x) = (x^2 + 2)/(x + 4) is the union of these intervals (,] and [,).\r\n" );
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document.write( "ANSWER. The range of the function  is the union of these intervals (,] and [,).\r\n" );
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document.write( "Solved.\r
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document.write( "It is how to solve this problem using only elementary Algebra and without using Calculus.\r
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document.write( "This problem is a typical Math Olympiad level problem or Math circle level problem
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document.write( "for 9-th grades high school students, who just know Algebra, but still don't know Calculus.\r
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