document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=132312>Question 132312</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having problems remembering domains.  I need to find the domain of the following:\r<BR>\n" );
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document.write( "f(t) = log(t-5)\r<BR>\n" );
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document.write( "Is the domain (-infinity, +infinity)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #96663 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=96663\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28t-5%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28t-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "Remember you cannot take the log of zero or of a negative value. So that means the argument <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t-5 ALIGN=MIDDLE  ALT=\"t-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be greater than zero (i.e. the argument <font size=4><b>must</b></font> be positive)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t-5%3E0 ALIGN=MIDDLE  ALT=\"t-5%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the inner expression greater than zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3E0%2B5 ALIGN=MIDDLE  ALT=\"t%3E0%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >Add 5 to both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3E5 ALIGN=MIDDLE  ALT=\"t%3E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side\r<BR>\n" );
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document.write( "So that means t must be greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > in order for t to be in the domain\r<BR>\n" );
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document.write( "So the domain in set-builder notation is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \textrm{\left{t|t>5\right}}\">\r<BR>\n" );
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document.write( "So here is the domain in interval notation: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(5,\infty\right)\">\r<BR>\n" );
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document.write( "Notice if we graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=log%2810%2C%28t-5%29%29 ALIGN=MIDDLE  ALT=\"y=log%2810%2C%28t-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (just replace t with x), we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+log%2810%2C%28x-5%29%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+log%2810%2C%28x-5%29%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > notice how the graph never crosses the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this graphically verifies our answer.\r<BR>\n" );
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document.write( "and we can see that x must be greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > in order to lie on the graph<BR>\n" );
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