document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1000359>Question 1000359</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Plz help me to explain the range of this equation. {f(x)=1÷x^2-2x-3} </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #617844 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=617844\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the denominator of that equation is x^2 - 2x - 3.\r<BR>\n" );
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document.write( "we'll call that g(x).\r<BR>\n" );
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document.write( "g(x) is a quadratic equation that can be solved to provide the following information:\r<BR>\n" );
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document.write( "the graph crosses the x-axis at x = -3 and x = 1.<BR>\n" );
document.write( "the minimum point on the graph is at (1,-4)<BR>\n" );
document.write( "the graph is positive from x > - infinity to x < -3.<BR>\n" );
document.write( "the graph is negative between x > -3 to x < 1.<BR>\n" );
document.write( "the graph goes positive again between x > 1 to x < infinity.\r<BR>\n" );
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document.write( "the range of g(x) is all values of y greater than or equal to -4.\r<BR>\n" );
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document.write( "f(x) is equal to 1 / g(x).\r<BR>\n" );
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document.write( "f(x) is therefore the reciprocal of g(x).\r<BR>\n" );
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document.write( "when g(x) is going towards positive infinity, f(x) is going towards 0 from the positive side.<BR>\n" );
document.write( "when g(x) is going towards negative infinity, f(x) is going towards 0 from the negative side.<BR>\n" );
document.write( "when g(x) is going towards 0 from the positive side, f(x) is going towards positive infinity.<BR>\n" );
document.write( "when g(x) is going towards 0 from the negative side, f(x) is going towards negative infinity.\r<BR>\n" );
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document.write( "every value of f(x) is going to be equal to 1 divided by every value of g(x).\r<BR>\n" );
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document.write( "when g(x) is at -4, f(x) will be at -.25.\r<BR>\n" );
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document.write( "the range of f(x) will be all values of y > 0 and all values of y <= -.25.\r<BR>\n" );
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document.write( "this can be seen on the graphs.\r<BR>\n" );
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document.write( "there are 3 graphs:\r<BR>\n" );
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document.write( "first graph is graph of g(x).<BR>\n" );
document.write( "second graph is graph of f(x).<BR>\n" );
document.write( "third graph is graph of f(x) and g(x) on the same graph.\r<BR>\n" );
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document.write( "in the third graph, the values of f(x) and g(x) are shown at x = -3, x = 1, and x = 5, so you can see that the values of f(x) are the reciprocal of the values of g(x) at those values of x.\r<BR>\n" );
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document.write( "this occurs at all values of x, but only those 3 values of x are shown for clarity.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/110101.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/110102.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/110103.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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