document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1027848>Question 1027848</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the functions<BR>\n" );
document.write( "f(x) = sqrt((x+1)/(x-1)) and g(x) = sqrt(x+1)/sqrt(x-1)<BR>\n" );
document.write( "Explain why f and g are not the same function. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #643056 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=643056\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28x%2B1%29%2F%28x-1%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28x%2B1%29%2F%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "g(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2B1%29%2Fsqrt%28x-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2B1%29%2Fsqrt%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "they look like the same function to me.\r<BR>\n" );
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document.write( "we'll graph both functions to see if that's true or not.\r<BR>\n" );
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document.write( "here's the graph of f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28x%2B1%29%2F%28x-1%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28x%2B1%29%2F%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/040302.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "here's the graph of g(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2B1%29%2Fsqrt%28x-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2B1%29%2Fsqrt%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/040303.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "as you can see, they are not the same function.\r<BR>\n" );
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document.write( "they are the same function when x is positive.\r<BR>\n" );
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document.write( "here's a spreadsheet of the data for x = -5 to x  +5.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/040305.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "to make a long story short, the basic difference is as follows:\r<BR>\n" );
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document.write( "with f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28x%2B1%29%2F%28x-1%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28x%2B1%29%2F%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the numerator and divisor are within the same square root sign.\r<BR>\n" );
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document.write( "if the numerator is negative and the denominator is negative, then the result is positive.\r<BR>\n" );
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document.write( "the square root of a positive number is valid, so the square root is allowed and it can be graphed.\r<BR>\n" );
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document.write( "with g(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2B1%29%2Fsqrt%28x-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2B1%29%2Fsqrt%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the numerator is under its own square root sign and the denominator is under its own square root sign.\r<BR>\n" );
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document.write( "if either the numerator is negative, or the denominator is negative, or botth are negative, then the result will be invalid because you can't take the square root of a negative number.\r<BR>\n" );
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document.write( "in general, square root of (a/b) is equal to square root of (a) / square root of (b).\r<BR>\n" );
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document.write( "this is true when both the numerator and denominator are positive.\r<BR>\n" );
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document.write( "when either the numerator or denominator or both are negative, then this is not true.\r<BR>\n" );
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document.write( "you then run into square root of a negative number, which is not valid.\r<BR>\n" );
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document.write( "if a is negative and b is positive, then:\r<BR>\n" );
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document.write( "sqrt(a/b) is not allowed because a/b is negative.\r<BR>\n" );
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document.write( "sqrt(a)/sqrt(b) is not allowed because a is negative.\r<BR>\n" );
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document.write( "likewise, if a is positive and b is negative, then:\r<BR>\n" );
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document.write( "sqrt(a/b) is not allowed because a/b is negative.\r<BR>\n" );
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document.write( "sqrt(a)/sqrt(b) is not allowed because b is negative.\r<BR>\n" );
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document.write( "if both a and b are negative, then:\r<BR>\n" );
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document.write( "sqrt(a/b) is allowed because a/b is positive.\r<BR>\n" );
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document.write( "sqrt(a)/sqrt(b) is not allowed because a is negative and because b is negative.\r<BR>\n" );
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