document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200475>Question 1200475</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If f(x) = {(2, 3), (4, 8), (7,  -  1)} and g(x) = {(8, 2), (  -  1, 4), (2, 7)}, <BR>\n" );
document.write( "find (f o g)(x), if it exists. </SPAN>\n" );
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document.write( "Answer: <font color=red size=4>(f o g)(x) = {(8,3), (-1,8), (2,-1)}</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "(f o g)(x) is the same as f( g(x) )\r<BR>\n" );
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document.write( "The g(x) function is<BR>\n" );
document.write( "g(x) = {(8,2), (-1,4), (2,7)}<BR>\n" );
document.write( "which means<BR>\n" );
document.write( "g(8) = 2<BR>\n" );
document.write( "g(-1) = 4<BR>\n" );
document.write( "g(2) = 7<BR>\n" );
document.write( "Each of those outputs (2,4 and 7) must be an input for the f(x) function to have f(g(x)) be possible.<BR>\n" );
document.write( "Luckily that is indeed the case.\r<BR>\n" );
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document.write( "f(x) = {(2,3), (4,8), (7,-1)}<BR>\n" );
document.write( "breaks down to<BR>\n" );
document.write( "f(2) = 3<BR>\n" );
document.write( "f(4) = 8<BR>\n" );
document.write( "f(7) = -1\r<BR>\n" );
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document.write( "Let's rewrite those three items above to involve g(x)<BR>\n" );
document.write( "f(2) = 3 ---> f(g(8)) = 3<BR>\n" );
document.write( "f(4) = 8 ---> f(g(-1)) = 8<BR>\n" );
document.write( "f(7) = -1 ---> f(g(2)) = -1\r<BR>\n" );
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document.write( "In short,<BR>\n" );
document.write( "f(g(8)) = 3<BR>\n" );
document.write( "f(g(-1)) = 8<BR>\n" );
document.write( "f(g(2)) = -1<BR>\n" );
document.write( "or<BR>\n" );
document.write( "(f o g)(8) = 3<BR>\n" );
document.write( "(f o g)(-1) = 8<BR>\n" );
document.write( "(f o g)(2) = -1<BR>\n" );
document.write( "Therefore,<BR>\n" );
document.write( "<font color=red size=4>(f o g)(x) = {(8,3), (-1,8), (2,-1)}</font>\r<BR>\n" );
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document.write( "Here's the input/output mapping diagram for each function piece. <BR>\n" );
document.write( "Start at the left side and move toward the right along the pathway of arrows shown.<BR>\n" );
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document.write( "For instance, if the input is x = 8, then<BR>\n" );
document.write( "g(8) = 2<BR>\n" );
document.write( "f(2) = 3 aka f(g(8)) = 3<BR>\n" );
document.write( "Therefore, f(g(x)) = 3 when the input is x = 8.\r<BR>\n" );
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document.write( "The g(x) function acts as the input to the f(x) function.<BR>\n" );
document.write( "Effectively g(x) takes on the role of x so to speak.<BR>\n" );
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