document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1000273>Question 1000273</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Plz explain the range of this equation...{1÷&#8730;x^2-2x-3=0} </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #617848 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=617848\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> this is the same as the last problem i answered except that the denominator is now the square root of (x^2 - 2x - 3) rather than just (x^2 - 2x - 3).\r<BR>\n" );
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document.write( "the equation in that problem was f(x) = 1 / (x^2 - 2x - 3).\r<BR>\n" );
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document.write( "the equation in this problem is f(x) = 1 / sqrt(x^2 - 2x - 3)).\r<BR>\n" );
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document.write( "that changes the range of f(x) because the domain is now different.\r<BR>\n" );
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document.write( "the equation we want to solve for is f(x).<BR>\n" );
document.write( "we'll call the equation inside the square root sign g(x).<BR>\n" );
document.write( "we'll call the square root of the equation inside the square root sign h(x).\r<BR>\n" );
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document.write( "what you get is:\r<BR>\n" );
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document.write( "f(x) = 1 / h(x), where h(x) = sqrt(g(x)) and g(x) = x^2 - 2x - 3.\r<BR>\n" );
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document.write( "put them all together and what you get is f(x) = 1 / sqrt(x^2 - 2x - 3).\r<BR>\n" );
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document.write( "we'll start with g(x) = x^2 - 2x - 3<BR>\n" );
document.write( "that can be factored to get g(x) = (x-3) * (x+1)<BR>\n" );
document.write( "the function g(x) is there equal to 0 when x = -1 and x = 3.<BR>\n" );
document.write( "the domain of g(x) is equal to all real values of x.<BR>\n" );
document.write( "there are no restrictions.<BR>\n" );
document.write( "the range of g(x) is equal to all real values of y >= -4.<BR>\n" );
document.write( "y has to be >= -4 beccause that is the minimum value of g(x).\r<BR>\n" );
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document.write( "the graph of g(x) is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/110101.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "now we'll go to h(x).\r<BR>\n" );
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document.write( "h(x) = sqrt(g(x)) = sqrt(x^2 - 2x - 3).<BR>\n" );
document.write( "after factoring of g(x), we get:<BR>\n" );
document.write( "h(x) = sqrt(g(x)) = sqrt((x-3)*(x+1)).<BR>\n" );
document.write( "h(x) will be positive if x > 3.<BR>\n" );
document.write( "h(x) will also be positive if x < -1, but that is not as clear as when x > 3.\r<BR>\n" );
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document.write( "let's look at what happens.\r<BR>\n" );
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document.write( "when x > 3, (x-3)*(x+1) becomes positive because both (x-3) and (x+1) are positive.\r<BR>\n" );
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document.write( "when x < -1, (x-3)*(x+1) becomes positive because both (x-3) and (x+1) are negative and a negative times a negative is positive.\r<BR>\n" );
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document.write( "between x > -1 and x < 3, the product will be negative because (x-3) will be negative and (x+1) will be positive and a negative times a positive is negative.\r<BR>\n" );
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document.write( "so your domain of h(x) will be all real values of x >= 3 and <= -1.<BR>\n" );
document.write( "the range of h(x) will be all real values of y >= 0.<BR>\n" );
document.write( "the range does not go down to -4 now because the values of x from x < 3 to x > -1 are not valid anymore because g(x) is now enclosed within the square root sign.\r<BR>\n" );
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document.write( "the graph of h(x) is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/110105.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "what happens to f(x)?\r<BR>\n" );
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document.write( "f(x) = 1 / h(x) which is equal to 1 / sqrt(x^2 - 2x - 3).\r<BR>\n" );
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document.write( "x can't be 3 and x can't be -1 because then the denominator will be equal to 0.<BR>\n" );
document.write( "x can't be > -1 and x can't be < 3 because than the denominator will not be real because of the square root function.\r<BR>\n" );
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document.write( "the domain of f(x) is therefore all real values of x < -1 and all real values of x > 3.\r<BR>\n" );
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document.write( "within those intervals, f(x) is the reciprocal of h(x) because f(x) = 1 / h(x) which is the same as f(x) = 1 / sqrt(x^2 - 2x - 3).\r<BR>\n" );
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document.write( "f(x) is therefore equal to the reciprocal of h(x) when x < -1 and when x > 3 only.\r<BR>\n" );
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document.write( "the graph of f(x) is shown below:\r<BR>\n" );
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document.write( "the range of f(x) is all real values of x > 0.\r<BR>\n" );
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document.write( "you don't have any negative values of f(x) because the domain of f(x) does not includ any values of x > -1 or < 3.\r<BR>\n" );
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document.write( "since the domain is not valid there, you can have no corresponding values of y within that interval.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/110106.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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