document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=216335>Question 216335</A>: <I><SPAN STYLE=\"word-break: break-all;\"> let f(x)=2x^2+3x-5 g(x)=4 solve for fog and gof </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #163492 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=163492\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E2%2B3x-5 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E2%2B3x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=4 ALIGN=MIDDLE  ALT=\"g%28x%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To solve for fog (pronounced \"f of g\") and gof (\"pronounced g of f\") it helps to understand function notation well. When you are given a formula (or rule) for a function (like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E2%2B3x-5 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E2%2B3x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >) it is important to understand the role of \"x\". It is simply a place-holder. It represents whatever input you provide to the function. And the rule illustrates what the particular function will do with its input in determining the output for that input.<br><BR>\n" );
document.write( "So when you see <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E2%2B3x-5 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E2%2B3x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > it shows you what that function f will do with its input, \"x\". It will square it, multiply that by 2, add 3 times the input and then subtract 5. The function f will do this to <b>any</b> input you give it.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E2%2B3x-5 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E2%2B3x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28q%29=2q%5E2%2B3q-5 ALIGN=MIDDLE  ALT=\"f%28q%29=2q%5E2%2B3q-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28z%29=2z%5E2%2B3z-5 ALIGN=MIDDLE  ALT=\"f%28z%29=2z%5E2%2B3z-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%289%29=2%289%29%5E2%2B3%289%29-5 ALIGN=MIDDLE  ALT=\"f%289%29=2%289%29%5E2%2B3%289%29-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%284x%2B3%29=2%284x%2B3%29%5E2%2B3%284x%2B3%29-5 ALIGN=MIDDLE  ALT=\"f%284x%2B3%29=2%284x%2B3%29%5E2%2B3%284x%2B3%29-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+4 ALIGN=MIDDLE  ALT=\"g%28x%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, since there is no \"x\" in the rule, the function g ignores the input! It simply returns a 4 no matter what input is given to the function!<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28y%29+=+4 ALIGN=MIDDLE  ALT=\"g%28y%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28q%29+=+4 ALIGN=MIDDLE  ALT=\"g%28q%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%5E2-3x%2B99%29+=+4 ALIGN=MIDDLE  ALT=\"g%28x%5E2-3x%2B99%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "etc.<BR>\n" );
document.write( "<br><BR>\n" );
document.write( "So now we are in position to figure out your problem. fog (\"f of g\") is another way of saying f(g(x)). Since g(x) = 4, f(g(x)) = f(4). And we saw above what f does to its input. It squares it, multiplies that by 2, etc. This is what it will do to 4:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%284%29=2%284%29%5E2%2B3%284%29-5 ALIGN=MIDDLE  ALT=\"f%284%29=2%284%29%5E2%2B3%284%29-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now all we have to do is go through the Order of Operations (aka PEMDAS) to simplify it:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%284%29=2%2816%29%2B3%284%29-5 ALIGN=MIDDLE  ALT=\"f%284%29=2%2816%29%2B3%284%29-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%284%29=32%2B12-5 ALIGN=MIDDLE  ALT=\"f%284%29=32%2B12-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%284%29=39 ALIGN=MIDDLE  ALT=\"f%284%29=39\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So fog = f(g(x)) = f(4) = 39<br><BR>\n" );
document.write( "gof (\"g of f\") is another way of saying g(f(x)). Now, as we saw above, the g function totally ignores its input and simply returns 4 <i>all the time</i>. So gof = g(f(x)) = 4!<BR>\n" );
document.write( " </SPAN>\n" );
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