document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208066>Question 1208066</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the integer function f(x)= int(2x), find A to D.\r<BR>\n" );
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document.write( "A. Find domain.\r<BR>\n" );
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document.write( "B. Find range.\r<BR>\n" );
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document.write( "C. Locate intercepts. \r<BR>\n" );
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document.write( "D. If f continuous on its domain?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846214 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52798)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=846214\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Given the integer function f(x)= int(2x), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28cross%28find%29%29 ALIGN=MIDDLE  ALT=\"highlight%28cross%28find%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <U>answer</U> A to D.\r<BR>\n" );
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document.write( "A. Find domain.<BR>\n" );
document.write( "B. Find range.<BR>\n" );
document.write( "C. Locate intercepts.<BR>\n" );
document.write( "D. If f continuous on its domain?<BR>\n" );
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document.write( "It is well known fact that the parent function int(x) is piecewise constant.\r\n" );
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document.write( "Making leaps at every integer point on x-axis, it takes a constant value of integer number n \r\n" );
document.write( "in each interval  n <= x < n+1  and is constant inside every such interval.\r\n" );
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document.write( "From it, it is clear that function f(x) = int(2x) is also piecewise constant.\r\n" );
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document.write( "Making leaps, it takes a constant value of integer number n in each half-integer interval \r\n" );
document.write( "over x-axis and is constant inside every such interval.\r\n" );
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document.write( "Same as the function int(x) is defined over the entire number line, function int(2x) \r\n" );
document.write( "is defined over the entire number line.  So, its domain is the set of all real numbers.\r\n" );
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document.write( "        Thus, question (A) is answered.\r\n" );
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document.write( "Its range is the set of all integer numbers (a discrete set).\r\n" );
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document.write( "        Thus, question (B) is answered, too.\r\n" );
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document.write( "Regarding intercepts.\r\n" );
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document.write( "y-intercept is  y = int(2*0)= int(0) = 0.  In other words, y-intercept is the point y= 0 at vertical y-axis x= 0,\r\n" );
document.write( "             or the point (0,0), the origin of the coordinate system.\r\n" );
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document.write( "x-intercept is the set of solution to equation y= 0.  \r\n" );
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document.write( "It is the set of points (x,0) on x-axis y= 0,  where  0 <= x < 1/2.\r\n" );
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document.write( "In other words, x-intercept is horizontal interval [0,1/2) on horizontal x-axis.\r\n" );
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document.write( "        So, question (C) is answered, too.\r\n" );
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document.write( "The answer to question (D) is obvious.\r\n" );
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document.write( "From what I said at the beginning, function f(x) = int(2x) is piecewise constant over every one half-a-unit interval\r\n" );
document.write( "on x-axis.  Such a function is INEVITABLY non-continuous. It fails/loses continuity at every half-integer point over x-axis.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved - - - all questions are answered.\r<BR>\n" );
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document.write( "==================\r<BR>\n" );
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document.write( "About parent function &nbsp;int(x), &nbsp;read this &nbsp;Wikipedia article\r<BR>\n" );
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document.write( "<A  rel=nofollow HREF=https://en.wikipedia.org/wiki/Floor_and_ceiling_functions>https://en.wikipedia.org/wiki/Floor_and_ceiling_functions</A>\r<BR>\n" );
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document.write( "https://en.wikipedia.org/wiki/Floor_and_ceiling_functions\r<BR>\n" );
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document.write( "The basic facts about this function from this article are the &nbsp;PRE-REQUISITES <BR>\n" );
document.write( "that you should know before you start solving this problem.\r<BR>\n" );
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document.write( "In this &nbsp;Wikipedia article, &nbsp;function &nbsp;int(x) &nbsp;is called &nbsp;\" a floor function \".\r<BR>\n" );
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document.write( "Happy learning &nbsp;(&nbsp;!&nbsp;)\r<BR>\n" );
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document.write( "<U>Comment from student</U>: I need to review the greatest integer function more carefully.\r<BR>\n" );
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document.write( "<U>My response</U> : &nbsp;&nbsp;If you need, &nbsp;do it.\r<BR>\n" );
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document.write( "From your comment, &nbsp;I did not get, &nbsp;what do you want from me, &nbsp;concretely and personally.\r<BR>\n" );
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document.write( "Also, &nbsp;I do not see your &nbsp;\" thanks \" &nbsp;in this comment, &nbsp;which puzzles me.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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