document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=126629>Question 126629</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help with function notation.  Not only do I not completely get the concept, I definitely don't understand graphing it.  Thanks so much\r<BR>\n" );
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document.write( "Graph f(x) = 3x + 2<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 #92780 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=92780\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm not sure whether you don't understand graphing, or you are having trouble because it says <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=3x%2B2 ALIGN=MIDDLE  ALT=\"f%28x%29=3x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > instead of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%2B2 ALIGN=MIDDLE  ALT=\"y=3x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Actually there is no difference between the two ways of expressing the relationship except to say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%2B2 ALIGN=MIDDLE  ALT=\"y=3x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > only guarantees that you have a relationship that makes the value of y dependent on the value of x.  (hence the terms dependent and independent variables).  The function notation on the other hand guarantees that the relationship is, in fact, a function.\r<BR>\n" );
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document.write( "So, what's the diff?  A relationship can be any rule that relates the two (or more) variables, but a function must have the property that for any given value of the independent variable (x in this case) for which the function is defined, there is one and only one value of the dependent variable (y) and therefore one and only one value for the function.  Another way to look at it is that if the relationship is a function, then there is no possible vertical line that intersects the graph of the function in more than one place.\r<BR>\n" );
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document.write( "As a counter-example consider this relationship: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2=4 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The graph of this relationship is a circle centered at the origin with a radius of 2.  With two exceptions, every vertical line that intersects this graph at all intersects it in two points.  Therefore, the relation is NOT a function.  On the other hand, the relation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sqrt%284-x%5E2%29 ALIGN=MIDDLE  ALT=\"y=sqrt%284-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is just the upper half of the same circle (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-sqrt%284-x%5E2%29 ALIGN=MIDDLE  ALT=\"y=-sqrt%284-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be the bottom half) IS a function because for all x in the domain of the function (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%3C=x%3C=2 ALIGN=MIDDLE  ALT=\"-2%3C=x%3C=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is one and only one value of the function. \r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%2B2 ALIGN=MIDDLE  ALT=\"y=3x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is, in fact, a function, we can write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=f%28x%29=3x%2B2 ALIGN=MIDDLE  ALT=\"y=f%28x%29=3x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "As to graphing when you are presented with a function in function notation, you can just rely on the equal relationship between y and f(x), or you can label the vertical axis on your graph as f(x) instead of the traditional y.\r<BR>\n" );
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document.write( "Since this is a 1st degree function (no term has an exponent greater than 1, and there are no xy terms), you should, by now, realize that your graph will be a straight line.  Straight lines are determined by no more and no less than two points.\r<BR>\n" );
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document.write( "All you have to do is pick a value for x and evaluate the function at the selected value to determine one of the points, which is then (x,f(x)).  Do this process twice, plot the points, and draw your line.\r<BR>\n" );
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document.write( "I usually like to pick values for x that make the arithmetic easy.  Let's use 0 and 1.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=3%280%29%2B2=2 ALIGN=MIDDLE  ALT=\"f%280%29=3%280%29%2B2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the first point is (0,2)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=3%281%29%2B2=5 ALIGN=MIDDLE  ALT=\"f%281%29=3%281%29%2B2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the second point is (1,5)\r<BR>\n" );
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