document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167698>Question 1167698</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How and what is the answer of the following?<BR>\n" );
document.write( " Mrs.cruz asks the class to graph the functions f(x) = x-2/x and g(x) = 3-2/x. Arnold and beth are asked to go to the chalkboard to graph f(x) = x-2/x and g(x) = 3- 2/x respectively.\r<BR>\n" );
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document.write( "a.Mrs.cruz asks Arnold and Beth to first writethedomain of each function. What would arnold write? What would beth write?\r<BR>\n" );
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document.write( "b.Mrs.cruz asks both students to determine the asymptotes of each function. What should each student say? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #792306 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=792306\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> arnold's equation is:\r<BR>\n" );
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document.write( "f(x) = x - 2/x\r<BR>\n" );
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document.write( "let y = f(x).\r<BR>\n" );
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document.write( "the equation becomes y = x - 2/x\r<BR>\n" );
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document.write( "place all terms under a common denominator to get:\r<BR>\n" );
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document.write( "y = (x^2 - 2)/x\r<BR>\n" );
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document.write( "the vertical asymptote is at x = 0.\r<BR>\n" );
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document.write( "since the degree of the numerator is 1 more than the degree of the denominator, the asymptote will be found by dividing the numerator by the denominator and using the quotient in the equation of the asymptote.\r<BR>\n" );
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document.write( "y = (x^2 - x) / x = x with a remainder of -2.\r<BR>\n" );
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document.write( "the quotient is x and the remainder is -2.\r<BR>\n" );
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document.write( "the equation of the quotient is y = x\r<BR>\n" );
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document.write( "the domain of the function is all values of x except at x = 0.\r<BR>\n" );
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document.write( "the range of the function is all values of y.\r<BR>\n" );
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document.write( "as x approaches 0 from the left, y approaches minus infinity.\r<BR>\n" );
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document.write( "as x approaches 0 from the right, y approaches plus infinity.\r<BR>\n" );
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document.write( "the slant asymptote is the line y = x.\r<BR>\n" );
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document.write( "this is what the graph looks like.\r<BR>\n" );
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document.write( "beth's equation is:\r<BR>\n" );
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document.write( "g(x) = 3 - 2/x\r<BR>\n" );
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document.write( "replace g(x) with y to get:\r<BR>\n" );
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document.write( "y = 3 - 2/x\r<BR>\n" );
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document.write( "place all terms under a common denominator to get:\r<BR>\n" );
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document.write( "y = (3x - 2) / x\r<BR>\n" );
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document.write( "the degree of the numerator is the same degree as the denominator, so there will be a horizontal asymptote.\r<BR>\n" );
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document.write( "divide 3x - 2 by x to get a quotient of 3 with a remainder of 2.\r<BR>\n" );
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document.write( "the equation of the asymptote will be y = 3\r<BR>\n" );
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document.write( "this is what the graph looks like.\r<BR>\n" );
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document.write( "both equations have a vertical asymptote at x = 0\r<BR>\n" );
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document.write( "therefore, the domain of both equations is all values of x except at x = 0.\r<BR>\n" );
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document.write( "the range of both functions is equal to all real values of y.\r<BR>\n" );
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document.write( "the fact that y is undefined at x = 0 doesn't affect the range of the equations.\r<BR>\n" );
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document.write( "there are no values of y that are not in the range of y .\r<BR>\n" );
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document.write( "in addition to the vertical asymptotes, .....\r<BR>\n" );
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document.write( "arnold's equation has a slant asymptote at y = x.\r<BR>\n" );
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document.write( "beth's equation has a horizontal asymptote at y = 3\r<BR>\n" );
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document.write( "here's a reference on horizontal and slant asymptotes.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.purplemath.com/modules/asymnote.htm\" target = \"_blank\">https://www.purplemath.com/modules/asymnote.htm</a> </SPAN>\n" );
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