document.write( "Question 89316: What is the domain and range of the following rational expression?\r
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Algebra.Com's Answer #65039 by Edwin McCravy(20059) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "What is the domain and range of the following \r\n" );
document.write( "rational expression? \r\n" );
document.write( "f(x)=x+3\r\n" );
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document.write( "x^2-5x-66\r\n" );
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document.write( "f(x) = \r\n" );
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document.write( "Factor the denominator:\r\n" );
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document.write( "f(x) = \r\n" );
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document.write( "The domain contains all numbers which \r\n" );
document.write( "can be substituted for x, and an answer\r\n" );
document.write( "is defined.  Denominators cannot be 0\r\n" );
document.write( "so x cannot be any number that will\r\n" );
document.write( "cause the denominator to be 0.\r\n" );
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document.write( "DENOMINATOR = (x-11)(x+2)\r\n" );
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document.write( "So x cannot equal to 11 or -2\r\n" );
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document.write( "So the domain is \r\n" );
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document.write( "or in interval notation (-,-2)U(-2,11)U(11,)\r\n" );
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document.write( "Now we have to find the range.\r\n" );
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document.write( "To do that we first replace f(x) by y and solve for x:\r\n" );
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document.write( "f(x) = \r\n" );
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document.write( "y = \r\n" );
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document.write( "Multiply both sides by the LCD = \r\n" );
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document.write( " = \r\n" );
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document.write( " = \r\n" );
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document.write( "Get 0 on the right:\r\n" );
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document.write( " = \r\n" );
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document.write( "Get the x terms together:\r\n" );
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document.write( " = \r\n" );
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document.write( "Factor x out of the second and third terms\r\n" );
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document.write( " = \r\n" );
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document.write( "factor -3 out of the last two terms on the left:\r\n" );
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document.write( " = \r\n" );
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document.write( "This is a quadratic equation in x with\r\n" );
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document.write( "a = y, b = -(5y+1), c = -3(22y+1)\r\n" );
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document.write( "We could solve for x, but we only\r\n" );
document.write( "need the discriminant b²-4ac to be\r\n" );
document.write( "non-negative, so\r\n" );
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document.write( "b²-4ac = [-(5y+1)]² - 4(y)[-3(22y+1)] =\r\n" );
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document.write( "25y² + 10y + 1 + 12y(22y+1) =\r\n" );
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document.write( "25y² + 10y + 1 + 264y² + 12y =\r\n" );
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document.write( "289y² + 22y + 1 > = 0\r\n" );
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document.write( "The left side has discriminant 22²-4(289)(1) = -672\r\n" );
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document.write( "thus it has no real zeros and represents a parabola which\r\n" );
document.write( "is always positive.\r\n" );
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document.write( "Therefore the range is \"all real numbers\" or (-,)\r\n" );
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document.write( "Edwin

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