document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185294>Question 1185294</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am a rational <BR>\n" );
document.write( "function having a vertical asymptote at <BR>\n" );
document.write( "the lines x = 3 and x = -3, and a <BR>\n" );
document.write( "horizontal asymptote at y = 1 . If my <BR>\n" );
document.write( "only x-intercept is 5, and my y-intercept <BR>\n" );
document.write( "is  -  5/9 , what function am I? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #816112 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=816112\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The conditions are over-specified; there is no rational function that satisfies all the conditions....<br><BR>\n" );
document.write( "(1) vertical asymptotes at x=3 and x=-3:<br><BR>\n" );
document.write( "This means there must be factor(s) of (x-3) and (x+3) in the denominator; and no other linear factors<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28x%29=a%2F%28%28%28x-3%29%5Em%29%28%28x%2B3%29%5En%29%29 ALIGN=MIDDLE  ALT=\"r%28x%29=a%2F%28%28%28x-3%29%5Em%29%28%28x%2B3%29%5En%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(2) horizontal asymptote at y=1:<br><BR>\n" );
document.write( "This means the leading terms of the numerator and denominator are the same (same coefficient and same power).  That means the coefficient a is 1, and the number of linear factors is the same in numerator and denominator.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28x%29=1%2F%28%28%28x-3%29%5Em%29%28%28x%2B3%29%5En%29%29 ALIGN=MIDDLE  ALT=\"r%28x%29=1%2F%28%28%28x-3%29%5Em%29%28%28x%2B3%29%5En%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(3) only x-intercept at x=5:<br><BR>\n" );
document.write( "The only factor(s) in the numerator are (x-5).  From (2), the total number of linear factors in the numerator and denominator must be the same<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28x%29=%28x-5%29%5E%28m%2Bn%29%2F%28%28%28x-3%29%5Em%29%28%28x%2B3%29%5En%29%29 ALIGN=MIDDLE  ALT=\"r%28x%29=%28x-5%29%5E%28m%2Bn%29%2F%28%28%28x-3%29%5Em%29%28%28x%2B3%29%5En%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(4) y-intercept -5/9:<br><BR>\n" );
document.write( "Set x=0 and see what happens<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%280%29=%28-5%29%5E%28m%2Bn%29%2F%28%28%28-3%29%5Em%29%28%283%29%5En%29%29=-5%2F9 ALIGN=MIDDLE  ALT=\"r%280%29=%28-5%29%5E%28m%2Bn%29%2F%28%28%28-3%29%5Em%29%28%283%29%5En%29%29=-5%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Ignoring signs, with only factors of 5 in the numerator and only factors of 3 in the denominator, the only way to get a y-intercept of 5/9 is with one factor of 5 in the numerator and two factors of 3 in the denominator.  But that would make the function<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28x%29=%28x-5%29%2F%28%28x-3%29%28x%2B3%29%29 ALIGN=MIDDLE  ALT=\"r%28x%29=%28x-5%29%2F%28%28x-3%29%28x%2B3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And that has both the wrong y-intercept (5/9 instead of -5/9) and the wrong horizontal asymptote (y=0 instead of y=1).<br><BR>\n" );
document.write( "ANSWER: There is no rational function with all the prescribed conditions<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );