document.write( "Question 1209186: what is the range of (fxg):
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Algebra.Com's Answer #847782 by ikleyn(52839) $\"\"$ $\"About$

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document.write( "The function (fxg)(x) is F(x) =  = 2x + .\r\n" );
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document.write( "They ask about the range of this function.\r\n" );
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document.write( "The plot is shown in this link  \r\n" );
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document.write( "    https://www.desmos.com/calculator/lh7qjcdhic \r\n" );
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document.write( "    https://www.desmos.com/calculator/lh7qjcdhic\r\n" );
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document.write( "There are two ways to solve this problem: one way is Algebra, and another way is Calculus.\r\n" );
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document.write( "              Algebra way\r\n" );
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document.write( "The range is the set of real values t such that\r\n" );
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document.write( "    2x +  = t  for some x.\r\n" );
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document.write( "In other words, the range is the set of real numbers t such that the quadratic equation\r\n" );
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document.write( "    2x^2 + 3 = tx,  or  2x^2 - tx + 3 = 0  \r\n" );
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document.write( "has real solutions for x.\r\n" );
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document.write( "For it, the necessary and sufficient condition is that the discriminant is non-negative\r\n" );
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document.write( "    d = (-t)^2 - 4*2*3 >= 0,  or  t^2 >= 24,   or  |t| >= ,  or\r\n" );
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document.write( "          +------------------------------------+\r\n" );
document.write( "          |    t <=    or  t >= .   |\r\n" );
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document.write( "Thus the range of the function  (fxg)  is the union of the two sets  \r\n" );
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document.write( "    (,] U [,).    ANSWER\r\n" );
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