document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183953>Question 1183953</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the domain and range of the function\r<BR>\n" );
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document.write( "f(x) = (x^2 + 1)/( 4x^2 - 8x + 3). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814466 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814466\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28x%5E2+%2B+1%29%2F%28+4x%5E2+-+8x+%2B+3%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28x%5E2+%2B+1%29%2F%28+4x%5E2+-+8x+%2B+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "domain:\r<BR>\n" );
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document.write( "exclude <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > that makes denominator equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+-+8x+%2B+3=0+ ALIGN=MIDDLE  ALT=\"4x%5E2+-+8x+%2B+3=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......use quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a ALIGN=MIDDLE  ALT=\"x=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-%28-8%29%2B-sqrt%28%28-8%29%5E2-4%2A4%2A3%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x=%28-%28-8%29%2B-sqrt%28%28-8%29%5E2-4%2A4%2A3%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%288%2B-sqrt%2864-48%29%29%2F8 ALIGN=MIDDLE  ALT=\"x=%288%2B-sqrt%2864-48%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%288%2B-sqrt%2816%29%29%2F8 ALIGN=MIDDLE  ALT=\"x=%288%2B-sqrt%2816%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%288%2B-4%29%2F8 ALIGN=MIDDLE  ALT=\"x=%288%2B-4%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%282%2B-1%29%2F2 ALIGN=MIDDLE  ALT=\"x=%282%2B-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "solutions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%282-1%29%2F2 ALIGN=MIDDLE  ALT=\"x=%282-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+1%2F2 ALIGN=MIDDLE  ALT=\"x=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%282%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"x=%282%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+3%2F2 ALIGN=MIDDLE  ALT=\"x=+3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, domain is:\r<BR>\n" );
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document.write( "{ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > element <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%3C%3E1%2F2 ALIGN=MIDDLE  ALT=\"+x%3C%3E1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E3%2F2+ ALIGN=MIDDLE  ALT=\"x%3C%3E3%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E3%2F2 ALIGN=MIDDLE  ALT=\"x%3E3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  }<BR>\n" );
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document.write( "Interval notation:\r<BR>\n" );
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document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ) U ( <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ) U ( <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity+ ALIGN=MIDDLE  ALT=\"infinity+\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "range:\r<BR>\n" );
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document.write( "the range is the set of y for which the discriminant is greater or equal to zero \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+%28x%5E2+%2B+1%29%2F%28+4x%5E2+-+8x+%2B+3%29 ALIGN=MIDDLE  ALT=\"y=+%28x%5E2+%2B+1%29%2F%28+4x%5E2+-+8x+%2B+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%28+4x%5E2-8x+%2B+3%29=+x%5E2+%2B+1 ALIGN=MIDDLE  ALT=\"y%28+4x%5E2-8x+%2B+3%29=+x%5E2+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4yx%5E2-x%5E2-+8yx+%2B+3y-1=0 ALIGN=MIDDLE  ALT=\"4yx%5E2-x%5E2-+8yx+%2B+3y-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284y-1%29x%5E2-+8yx+%2B+%283y-1%29=0 ALIGN=MIDDLE  ALT=\"%284y-1%29x%5E2-+8yx+%2B+%283y-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "->discriminant >=0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-8y%29%5E2-4%284y-1%29%283y-1%29 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-8y%29%5E2-4%284y-1%29%283y-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=64y%5E2-48+y%5E2+%2B+28+y+-+4 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=64y%5E2-48+y%5E2+%2B+28+y+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=16y%5E2+%2B+28+y+-+4 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=16y%5E2+%2B+28+y+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16y%5E2+%2B+28+y+-+4+%3E=0 ALIGN=MIDDLE  ALT=\"16y%5E2+%2B+28+y+-+4+%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .......divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2+%2B+7y+-+1+%3E=0 ALIGN=MIDDLE  ALT=\"4y%5E2+%2B+7y+-+1+%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "use quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"y=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-7%2B-sqrt%287%5E2-4%2A4%28-1%29%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"y=%28-7%2B-sqrt%287%5E2-4%2A4%28-1%29%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-7%2B-sqrt%2849%2B16%29%29%2F8 ALIGN=MIDDLE  ALT=\"y=%28-7%2B-sqrt%2849%2B16%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-7%2B-sqrt%2865%29%29%2F8 ALIGN=MIDDLE  ALT=\"y=%28-7%2B-sqrt%2865%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "solutions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2865%29-7%29%2F8%3E=0+ ALIGN=MIDDLE  ALT=\"%28sqrt%2865%29-7%29%2F8%3E=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-7-sqrt%2865%29%29%2F8%3E=0+ ALIGN=MIDDLE  ALT=\"%28-7-sqrt%2865%29%29%2F8%3E=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "range is:\r<BR>\n" );
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document.write( "{  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f+%28x+%29 ALIGN=MIDDLE  ALT=\"f+%28x+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > element <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > :  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f+%28x+%29+%3C=%28-7-sqrt%2865%29%29%2F8 ALIGN=MIDDLE  ALT=\"f+%28x+%29+%3C=%28-7-sqrt%2865%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f+%28x+%29+%3E=%28sqrt%2865%29-7%29%2F8 ALIGN=MIDDLE  ALT=\"+f+%28x+%29+%3E=%28sqrt%2865%29-7%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > }\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );