document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=557449>Question 557449</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Range of function sqrt(x^2-2x+5). Why is the range &#8805;2?, without graphing </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #362610 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=362610\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i cheated by graphing it.<BR>\n" );
document.write( "now i see what's going on.<BR>\n" );
document.write( "what is happening is that the range of the function will be >= 2 because the value of y will never get below 2.<BR>\n" );
document.write( "the domain of this function is all values of x such that the sqrt(x^2-2x+5) is positive.<BR>\n" );
document.write( "that will occur at all values of x.<BR>\n" );
document.write( "when x = -5, the value of y is equal to 6.3245553<BR>\n" );
document.write( "when x = 0, the value of y is equal to 2.236068<BR>\n" );
document.write( "when x = 5, the value of y is equal to 4.472136<BR>\n" );
document.write( "the minimum value of y occurs when x = = 1<BR>\n" );
document.write( "y never gets below 2.<BR>\n" );
document.write( "y can go as high as you want, depending on the value of x.<BR>\n" );
document.write( "for example:<BR>\n" );
document.write( "when x = -500, y = 501.00399<BR>\n" );
document.write( "when x = 500, y = 499.00401<BR>\n" );
document.write( "when x = -5000, y = 5001.0004<BR>\n" );
document.write( "when x = 5000, y = 49999.0004<BR>\n" );
document.write( "the greater the value of x, the closer y will get to the absolute value of x.<BR>\n" );
document.write( "so, if x can go infinite in each direction, this means that y can go infinite in the positive direction, because y will always be positive.<BR>\n" );
document.write( "on the reverse side, there is no value of x that will provide a value of y smaller than 2.<BR>\n" );
document.write( "that is why the range of the function is >= 2.<BR>\n" );
document.write( "we are talking about the range of the function which is the value of y or is the value of f(x) which is equal to y.<BR>\n" );
document.write( "the domain is the value of x which can be any real number.<BR>\n" );
document.write( "the graph of this equation, by the way, looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-5%2C5%2C-5%2C5%2Csqrt%28x%5E2-2x%2B5%29%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-5%2C5%2C-5%2C5%2Csqrt%28x%5E2-2x%2B5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "as x gets more negative and as x gets more positive, the value of y will get more positive.<BR>\n" );
document.write( " </SPAN>\n" );
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