document.write( "Question 1188782: Find the domain of each function. Find any horizontal, vertical, or oblique asymptotes.
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Algebra.Com's Answer #819957 by Edwin McCravy(20064)\"\" \"About 
You can put this solution on YOUR website!
Instead of doing your homework for you I will do problems with different numbers \r\n" );
document.write( "that are done step-by-step as yours are done, so you can use them as a step-by-step \r\n" );
document.write( "model to do yours by.
(a) R(x)= (x-5)/(x+6)^2
The denominator cannot equal 0, so set (x+6)^2 = 0 and solve, getting x = -6,\r\n" );
document.write( "so this tells you everything that x cannot equal so x cannot be -6, so the domain\r\n" );
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document.write( "and there is a vertical asymptote at x=-6.\r\n" );
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document.write( "The denominator has a greater degree than the numerator, so the horizontal\r\n" );
document.write( "asymptote is the x-axis, which has equation y=0.

\n" ); document.write( "(b) f(x)= (3x^2-33x+90)/(x^2+3x-54) = (3(x^2-11x+30))/((x+9)(x-6))
The denominator cannot equal 0, so set (x+9)(x-6) = 0 and solve, getting \r\n" );
document.write( "x = -9, and x = 6. This tells you everything that x cannot equal, so x cannot be\r\n" );
document.write( "-9 or 6, so the domain\r\n" );
document.write( "is \r\n" );
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document.write( "To find the asymptotes and/or holes, we first must factor both numerator and\r\n" );
document.write( "denominator to see if we have any holes (because of like factors in numerator\r\n" );
document.write( "and denominator).

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document.write( "The fact that (x-6) is a like factor of both numerator and denominator means\r\n" );
document.write( "that there is a hole, not a vertical asymptote, at x=6.  So the only vertical\r\n" );
document.write( "asymptote is when x+9 = 0 or at x = -9.\r\n" );
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document.write( "Since the degrees of the numerator and the denominator are both 2, (x2),\r\n" );
document.write( "the equation of the horizontal asymptote is found by setting y\r\n" );
document.write( "equal to the fraction with the numerator equal to the leading coefficient of the\r\n" );
document.write( "numerator and the denominator equal to the leading coefficient of the\r\n" );
document.write( "denominator.  So the horizontal asymptote has equation y = 3/1 or y = 3.\r\n" );
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(c) g(x)= (x^2+5x-14)/(x+2)
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document.write( "The denominator cannot equal 0, so set (x+2) = 0 and solve, getting x = -2, so\r\n" );
document.write( "this tells you everything that x cannot equal so x cannot be -1, so the domain\r\n" );
document.write( "is \r\n" );
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document.write( "To find the asymptotes and/or holes, we first must factor the numerator to see\r\n" );
document.write( "if we have any holes (because of like factors in numerator and denominator).\r\n" );
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document.write( "The fact that there are no like factors of both numerator and denominator means\r\n" );
document.write( "that there is no hole when x+2=0, but a vertical asymptote.  So the only\r\n" );
document.write( "vertical asymptote is when x+2 = 0 or at x = -2.\r\n" );
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document.write( "Since the degree of the numerator is 2 and the degree of the denominator is 1,\r\n" );
document.write( "that is, the numerator's degree is 1 more than the denominator's degree, that\r\n" );
document.write( "means there is an oblique asymptote.  To find its equation, divide the\r\n" );
document.write( "numerator by the denominator using long division (or synthetic division if you\r\n" );
document.write( "understand the significant parts of synthetic division.  I will use long\r\n" );
document.write( "division in case you don't:\r\n" );
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document.write( "        x+ 3\r\n" );
document.write( "x+2)x2+5x-14\r\n" );
document.write( "    x2+2x\r\n" );
document.write( "       3x-14\r\n" );
document.write( "       3x+ 6\r\n" );
document.write( "         -20\r\n" );
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document.write( "So the quotient is x+3-20/(x+2).  We ignore the fraction because it will\r\n" );
document.write( "approach 0 as x gets very large positively and negatively. So the equation of\r\n" );
document.write( "the oblique asymptote is gotten by setting y equal to just the quotient.\r\n" );
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document.write( "Oblique asymptote is y = x+3   \r\n" );
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document.write( "Now use these as a model and do yours step-by-step exactly the same way.  \r\n" );
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document.write( "Edwin
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