document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=884013>Question 884013</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In Fantasy Island an individual pays 10% tax on gross annual income not exceeding $200,000.  For every dollar on the next $300,000 the tax is 15%.  Any income in excess of this is taxed at 20 cents in the dollar.<BR>\n" );
document.write( "Using T to represent annual tax liability and x gross annual income:  <BR>\n" );
document.write( "a.   Determine the annual tax liability function T(x) for an income earner in Fantasy Island.              <BR>\n" );
document.write( "b.   What is the tax liability of a person who earns a gross annual income of $350,000?       <BR>\n" );
document.write( "c.   What is the tax liability of a person who earns a gross annual income of $500,000?     <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 #533957 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=533957\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> very tricky function and i just figured it out after many previous tries where i failed.\r<BR>\n" );
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document.write( "you are the beneficiary of this sudden breakthrough in knowledge.\r<BR>\n" );
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document.write( "previously i thought it had to be a piecewise function, which it can be.\r<BR>\n" );
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document.write( "today i learned that it doesn't have to be, and can be modeled with one continuous function.\r<BR>\n" );
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document.write( "i'm not sure if you're supposed to be sophisticated enough to figure this out, but i'll give you the solution and you can certainly use it if you like.\r<BR>\n" );
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document.write( "If you can't use it, or don't want to, then you'll need to resort to a piecewise function or figure out another function on your own.\r<BR>\n" );
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document.write( "the function that i just discovered today is shown in the following pictures along with examples of how it would be used.\r<BR>\n" );
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document.write( "the formula uses (x-0) + |x-0| and (x-100) + |x-200| and (x-500) + |x-500|\r<BR>\n" );
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document.write( "the expression |...| means absolute value of (...).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/jun2403_0001.jpg\" alt=\"$$$\" </><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/jun2403_0002.jpg\" alt=\"$$$\" </><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/jun2403_0003.jpg\" alt=\"$$$\" </><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/jun2403_0004.jpg\" alt=\"$$$\" </><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/jun2403_0005.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the graph of the equation looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/jun2404_0002.jpg\" alt\"$$$\" </>\r<BR>\n" );
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document.write( "you can confirm the solution is accurate by using the folloiwng logic.\r<BR>\n" );
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document.write( "x = 200<BR>\n" );
document.write( "tax = .10 * 200 = 20.\r<BR>\n" );
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document.write( "x = 350<BR>\n" );
document.write( "tax = .10 * 200 + .15 * 150 = 20 + 22.5 = 42.5\r<BR>\n" );
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document.write( "x = 500<BR>\n" );
document.write( "tax = .10 * 200 + .15 * 300 = 20 + 45 = 65\r<BR>\n" );
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document.write( "x = 800<BR>\n" );
document.write( "tax = .10 * 200 + .15 * .15 * 300 + .20 * 300 = 20 + 45 + 60 = 125\r<BR>\n" );
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document.write( "this is because:\r<BR>\n" );
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document.write( "you pay 10% for the amount above 0 and less than or equal to 200.<BR>\n" );
document.write( "you pay 15% for the amount above 200 and less than or equal to 500.<BR>\n" );
document.write( "you pay 20% for the amount over 500.\r<BR>\n" );
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document.write( "the expression works because:\r<BR>\n" );
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document.write( "when x is smaller than y, the expression (x-y) becomes -z and the expression |x-y| becomes + z and the -z and the +z cancel each other out resulting in 0.\r<BR>\n" );
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document.write( "when x is larger than y, the expression (x-y) becomes z and the expression |x-y| becomes z and you get 2z which is then equal to z after you divide it by 2.\r<BR>\n" );
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