document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=230026>Question 230026</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is the equation a function? Why or why not?<BR>\n" );
document.write( "x^2+x^3+y=6x<BR>\n" );
document.write( "I put it in slope-intercept form like so: y=6x-x^5\r<BR>\n" );
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document.write( "Also, is this a function, why or why not?:<BR>\n" );
document.write( "y^2+2x-4=o\r<BR>\n" );
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document.write( "Thank you for your time </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #170438 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=170438\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "NO! NO! NO!  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  x^2 + x^3 \neq x^5\">  No how, no way.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  (x^2)(x^3) = x^5\"> is a true statement.  Also, you can't put a cubic into 'slope-intercept' form.  Slope-intercept is a form of <u>linear</u> equation, and this is anything but linear.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2 + x^3 + y\ =\ 6x\ \rightarrow\ y\ =\ -x^3-x^2+6x\">\r<BR>\n" );
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document.write( "Is it a function?\r<BR>\n" );
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document.write( "Well, it is a relation that maps values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> to values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">.  The question is, is there any value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> that maps to more than one value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">?\r<BR>\n" );
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document.write( "In general, all polynomial relations of the form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ a_nx^n\ +\ a_{n-1}x^{n-1}\ +\cdots+\ a_0\"> are functions.\r<BR>\n" );
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document.write( "Let's examine a graph of your relation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28%0D%0A500%2C+500%2C+-5%2C+5%2C+-10%2C+10%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28%0D%0A500%2C+500%2C+-5%2C+5%2C+-10%2C+10%2C%0D%0A-x%5E3-x%5E2%2B6x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The test to see if you have a function is the vertical line test.  If you can find a vertical line that intersects the graph of the relation in more than one point, then you do NOT have a function.  In the case of the given relation, there is no such vertical line, therefore the relation is, in fact, a function.\r<BR>\n" );
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document.write( "On the other hand,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y^2\ +\ 2x\ -\ 4\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y^2\ = -2x\ +\ 4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ \pm\sqrt{-2x\ +\ 4}\">\r<BR>\n" );
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document.write( "Let's look at this graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28%0D%0A500%2C+500%2C+-5%2C+5%2C+-5%2C+5%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28%0D%0A500%2C+500%2C+-5%2C+5%2C+-5%2C+5%2C%0D%0Asqrt%28-2x%2B4%29%2C%0D%0A-sqrt%28-2x%2B4%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Oops!  Lots of vertical lines that intersect the graph in two places, in fact, there are an infinity of them.  Also, there is only one vertical line that intersects the graph in exactly one point, and that would be the vertical line that is tangent to the curve at the vertex, the vertical line <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x=2\">.\r<BR>\n" );
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document.write( "So, NOT a function.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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