document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196851>Question 1196851</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the domain and range of f(x) =(x+1)^(x-1) <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829915 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=829915\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Clearly x = -1 is a critical value in determining the domain and range.<br><BR>\n" );
document.write( "At x = -1, the function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2B1%29%5E%28-1-1%29=0%5E%28-2%29=1%2F0%5E2 ALIGN=MIDDLE  ALT=\"%28-1%2B1%29%5E%28-1-1%29=0%5E%28-2%29=1%2F0%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "which is undefined.<br><BR>\n" );
document.write( "So x = -1 is not in the domain.<br><BR>\n" );
document.write( "For any value of x greater than -1, the expression is a power of a positive number, so the function value can always be evaluated.  So (-1, infinity) is at least part of the domain.<br><BR>\n" );
document.write( "The tricky part of determining the domain is with values of x less than -1.  That results in a negative number raise to a negative power.<br><BR>\n" );
document.write( "A negative number raised to a negative power is a real number if and only if the power is an integer.  For example...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-2%29=%28-1%29%5E%28-3%29+=+1%2F%28-1%29%5E3+=+-1 ALIGN=MIDDLE  ALT=\"f%28-2%29=%28-1%29%5E%28-3%29+=+1%2F%28-1%29%5E3+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29=%28-2%29%5E%28-4%29=1%2F%28-2%29%5E4=1%2F16+=+0.625 ALIGN=MIDDLE  ALT=\"f%28-3%29=%28-2%29%5E%28-4%29=1%2F%28-2%29%5E4=1%2F16+=+0.625\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-4%29=%28-3%29%5E%28-5%29=1%2F%28-3%29%5E5=-1%2F243=-0.0041 ALIGN=MIDDLE  ALT=\"f%28-4%29=%28-3%29%5E%28-5%29=1%2F%28-3%29%5E5=-1%2F243=-0.0041\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "But for non-integer values of x less than -1, the function value is not a real number.<br><BR>\n" );
document.write( "So....<br><BR>\n" );
document.write( "Depending on your definition of domain, the domain of the function is either<br><BR>\n" );
document.write( "(-1,infinity)<br><BR>\n" );
document.write( "or<br><BR>\n" );
document.write( "(-1,infinity) U {integers less than -1}<br><BR>\n" );
document.write( "For the range, unless you know some very high-level mathematics, you can use a graphing calculator to find the answer.  It should be clear that there is no upper limit to the range; my TI-83 calculator shows a minimum value of approximately 0.815154 at approximately x=0.454731.<br><BR>\n" );
document.write( "So the range is approximately (0.815154,infinity).<br><BR>\n" );
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