document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=443978>Question 443978</A>: <I><SPAN STYLE=\"word-break: break-all;\"> My teacher has told me that it is impossible to find a domain for equations like:\r<BR>\n" );
document.write( "\n" );
document.write( "sqrt(-1-2x)<BR>\n" );
document.write( "sqrt(-2x-4)\r<BR>\n" );
document.write( "\n" );
document.write( "I can't understand why the domain would not be (-&#8734;, -½) and (-&#8734;, -2).  I was told that finding a domain was impossible because both elements in the equation were negative (-1 and -2x; -2x and -4).  However, because x is an unknown and could be negative, I don't quite see how what I was told is possible.  Is the variable x not considered when finding domains? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #306158 by </B><A HREF=/tutors/aboutme.mpl?userid=swincher4391><B>swincher4391(1107)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=swincher4391><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=swincher4391><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=306158\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It doesn't sound like your teach knows what they're talking about.\r<BR>\n" );
document.write( "\n" );
document.write( "There are multiple ways to find the domains of these functions:\r<BR>\n" );
document.write( "\n" );
document.write( "One find the inverse of the function's range (which is the original's domain)<BR>\n" );
document.write( "Two solve for where x is defined.\r<BR>\n" );
document.write( "\n" );
document.write( "===============<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1-2x%29 ALIGN=MIDDLE  ALT=\"sqrt%28-1-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > it is defined strictly where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1-2x+%3E=+0 ALIGN=MIDDLE  ALT=\"-1-2x+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So solve for x to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+%3E=+2x ALIGN=MIDDLE  ALT=\"-1+%3E=+2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2%3E=x%7D%7D+%7B%7B%7Bx%3C=+-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2%3E=x%7D%7D+%7B%7B%7Bx%3C=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From -<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <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;\" > is the domain.<BR>\n" );
document.write( "===============<BR>\n" );
document.write( "You can find the inverse as well.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%28-1-2x%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%28-1-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+=+-1-2x ALIGN=MIDDLE  ALT=\"y%5E2+=+-1-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+%2B+1+=+-2x ALIGN=MIDDLE  ALT=\"y%5E2+%2B+1+=+-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2A%28y%5E2%2B1%29%2F2+=+x ALIGN=MIDDLE  ALT=\"-1%2A%28y%5E2%2B1%29%2F2+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Switch the xs and ys to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+expr%28-1%2F2%29%28x%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"y+=+expr%28-1%2F2%29%28x%5E2%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the range of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=1 ALIGN=MIDDLE  ALT=\"x%3E=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > then if you multiplied all values by -1/2, you'd get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=-1%2F2 ALIGN=MIDDLE  ALT=\"x%3C=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the same as what we had.<BR>\n" );
document.write( "===================<BR>\n" );
document.write( "I believe the argument you teacher may have tried to make is:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1-2x%29+=+sqrt%28-1%2A%281%2B2x%29%29+=+sqrt%28-1%29+%2A+sqrt%281%2B2x%29 ALIGN=MIDDLE  ALT=\"sqrt%28-1-2x%29+=+sqrt%28-1%2A%281%2B2x%29%29+=+sqrt%28-1%29+%2A+sqrt%281%2B2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not allowed, then you can't find the domain. But this is not true. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1%2A%281%2B2x%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28-1%2A%281%2B2x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is defined where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B2x%29%3C=+0 ALIGN=MIDDLE  ALT=\"%281%2B2x%29%3C=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, because a negative times a negative gives us a positive (which we can take the square root of). So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B2x%29+%3C=+0 ALIGN=MIDDLE  ALT=\"%281%2B2x%29+%3C=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1+%3C+-2x ALIGN=MIDDLE  ALT=\"+1+%3C+-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2+%3E=+x ALIGN=MIDDLE  ALT=\"-1%2F2+%3E=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+-1%2F2 ALIGN=MIDDLE  ALT=\"x+%3C=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >... again gives us the same result.<BR>\n" );
document.write( "================<BR>\n" );
document.write( "Thus you can find the domain. <BR>\n" );
document.write( "===============<BR>\n" );
document.write( "Does this make sense? I hope this helped. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );