document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=243395>Question 243395</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Trting to assist 13 yr old son...I know very little about algebra.\r<BR>\n" );
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document.write( "Find the domain of this rational function.\r<BR>\n" );
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document.write( "1)f(x)=5x+6/2x-6\r<BR>\n" );
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document.write( "2)f(x)=3x-8/14x+20\r<BR>\n" );
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document.write( "Thanks for any assistance\r<BR>\n" );
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document.write( "Christine </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #178347 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=178347\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> PROBLEM NUMBER 1\r<BR>\n" );
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document.write( "f(x)=5x+6/2x-6\r<BR>\n" );
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document.write( "The domain of function number 1 is all real values of x except where the denominator is equal to 0.\r<BR>\n" );
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document.write( "The denominator is equal to 0 when (2x-6) = 0\r<BR>\n" );
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document.write( "Solve for x in 2x-6 = 0<BR>\n" );
document.write( "add 6 to both sides to get 2x = 6<BR>\n" );
document.write( "divide both sides by 2 to get x = 3\r<BR>\n" );
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document.write( "the domain becomaes all real values of x except x = 3.\r<BR>\n" );
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document.write( "the range is all real values of y because there does not appear to be any restriction on the values of y.\r<BR>\n" );
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document.write( "a graph of this equation looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28600%2C600%2C-10%2C10%2C-50%2C50%2C%285x%2B6%29%2F%282x-6%29%29 ALIGN=MIDDLE  ALT=\"graph+%28600%2C600%2C-10%2C10%2C-50%2C50%2C%285x%2B6%29%2F%282x-6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The equation has a vertical asymptote at x = 3.\r<BR>\n" );
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document.write( "The value of y approaches negative infinity as x approaches 3 from below 3.\r<BR>\n" );
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document.write( "the value of y approaches positive infinity as x approaches 3 from above 3.\r<BR>\n" );
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document.write( "This particular equation looks like it has a horizontal asymptote at y = 5/2.\r<BR>\n" );
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document.write( "the following graph might show that clearer.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28600%2C600%2C-100%2C100%2C-50%2C50%2C%285x%2B6%29%2F%282x-6%29%2C5%2F2%29 ALIGN=MIDDLE  ALT=\"graph+%28600%2C600%2C-100%2C100%2C-50%2C50%2C%285x%2B6%29%2F%282x-6%29%2C5%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the horizontal line at 5/2 shows that the horizontal asymptote appears to be at y = 5/2.\r<BR>\n" );
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document.write( "But, you were not asked to show that.\r<BR>\n" );
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document.write( "You were only asked to find the domain.\r<BR>\n" );
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document.write( "The domain is all real values of x except at x = 3.\r<BR>\n" );
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document.write( "PROBLEM NUMBER 2\r<BR>\n" );
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document.write( "f(x)=3x-8/14x+20\r<BR>\n" );
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document.write( "It looks like the domain here is restricted by when the denominator is equal to 0 again.\r<BR>\n" );
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document.write( "The denominator is equal to 0 when 14x + 20 is equal to 0.\r<BR>\n" );
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document.write( "14x + 20 is equal to 0 when x = ???\r<BR>\n" );
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document.write( "We have to solve for x when 14x + 20 = 0\r<BR>\n" );
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document.write( "14x + 20 = 0<BR>\n" );
document.write( "subtract 20 from both sides to get 14x = -20<BR>\n" );
document.write( "divide both sides by 14 to get x = -20/14 = -10/7\r<BR>\n" );
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document.write( "It looks like the domain will be all real values of x except when x = -10/7\r<BR>\n" );
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document.write( "The two biggies for restricting the domain are square root of a negative number and division by 0.\r<BR>\n" );
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document.write( "This problem deals with division of 0 only.\r<BR>\n" );
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document.write( "A graph of this equation looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-10%2C10%2C-20%2C20%2C%283x-8%29%2F%2814x%2B20%29%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-10%2C10%2C-20%2C20%2C%283x-8%29%2F%2814x%2B20%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "It's hard to see, but there is a vertical asymptote at x = -10/7 which is roughly equivalent to -1 and - 3/7, or -1.428571429.\r<BR>\n" );
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document.write( "This graph also looks like it has a horizontal asymptote at y = 3/14 but, again, you don't need to worry about that for the questions you are being asked.\r<BR>\n" );
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document.write( "If you need to brush up on horizontal or vertical asymptotes, the following website might be helpful.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.ehow.com/how_4897380_vertical-horizontal-asymptotes.html\" target = \"_blank\">Horizontal and Vertical Asymptores</a>\r<BR>\n" );
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document.write( "Vertical asymptotes are fairly easy.\r<BR>\n" );
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document.write( "You just have to find when the denominator is equal to 0.\r<BR>\n" );
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document.write( "Horizontal asymptotes are a little more difficult because you have to take the limit of x as x approaches infinity.\r<BR>\n" );
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document.write( "In your first equation, what I did was this:\r<BR>\n" );
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document.write( "5x+6/2x-6 becomes 5x/2x as x approaches infinity because the 6 and the -6 become insignificant as x gets very large.\r<BR>\n" );
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document.write( "5x/2x becomes 5/2 because the x cancels out.\r<BR>\n" );
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document.write( "I did a similar procedure for your second equation.\r<BR>\n" );
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document.write( "Your domain, however, was a different question.\r<BR>\n" );
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document.write( "It was what value of x can be used.\r<BR>\n" );
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document.write( "In general, all real values of x can be used except where the equation is not defined for real values of x.\r<BR>\n" );
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document.write( "The equation is not defined when the denominator is equal to 0.\r<BR>\n" );
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document.write( "The equation is not defined when you wind up with the square root of a negative number.\r<BR>\n" );
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document.write( "Those are the two most common causes.\r<BR>\n" );
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document.write( "Consider y = square root of x-3\r<BR>\n" );
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document.write( "when x is smaller than 3, the square root becomes negative which is not a real number so the domain is restricted to values of x greater than or equal to 3.\r<BR>\n" );
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document.write( "a graph of this equation would look like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-3%2C10%2C-3%2C10%2Csqrt%28x-3%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-3%2C10%2C-3%2C10%2Csqrt%28x-3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "you can see that the graph is not defined for any values of x smaller than 3.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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