document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=181302>Question 181302</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Are all linear equations functions?I would say no but In what instance is a linear equation not a function?I need an equation of nonlinear function and I don't knwo how to write that or where to start. Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #135940 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=135940\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a <i><b>single output</b></i> to each input element drawn from a fixed set, such as the real numbers, although different inputs may have the same output.\r<BR>\n" );
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document.write( "This definition leads directly to what is called the Vertical Line Test.  If you graph a mathematical relationship in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large R^2\"> and you can find <i><b>any</b></i> vertical line that intersects the graph in more than one point, then the relation is <i><b>not</b></i> a function.\r<BR>\n" );
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document.write( "All two-variable linear equations graph to straight lines, so the only graphs that would be intersected in more than one point by a vertical line would be the graph of a vertical line.  Hence, all linear equations <i><b>except</b></i> for the sub-set of linear equations of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = a\"> are functions.  An equation of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = a\"> would be intersected by the vertical line <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = a\"> in <i><b>all</b></i> of its points, causing the Vertical Line Test to fail.\r<BR>\n" );
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document.write( "So, the answer to your initial question is no.  All of the instances where a linear equation mapped to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large R^2\"> is <i><b>not</b></i> a function is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \{Ax + By = C | x,\  y,\  A, \ C\  \in \R,\  A \neq 0,\  B = 0\}\">.\r<BR>\n" );
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document.write( "A non-linear equation is anything that doesn't graph to a straight line.  The equation of a non-linear <b>function</b> must also pass the Vertical Line Test.  Common examples are polynomial equations of degree <i>n</i> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n > 1\">, exponential equations such as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b^x = y\">, or logarithmic equations such as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log_b{x} = y\">.\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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