document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1160675>Question 1160675</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find functions f and g such that h= g o f. (Note: The answer is not unique.<BR>\n" );
document.write( "Enter your answers as a comma-separated list of functions. Use non-identity<BR>\n" );
document.write( "functions for f and g.)\r<BR>\n" );
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document.write( "h(x)=(5x-3)^3/2\r<BR>\n" );
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document.write( "{g(x), f(x)} =  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #784051 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=784051\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let me expand on the answer from tutor @Edwin, which shows one pair of functions that satisfy the conditions of the problem.<br><BR>\n" );
document.write( "As your post says, the answer is not unique; there are other possible answers for the functions f and g.<br><BR>\n" );
document.write( "Consider the sequence of operations the function h(x) does to the input value x:<BR>\n" );
document.write( "(1) multiply by 5<BR>\n" );
document.write( "(2) subtract 3<BR>\n" );
document.write( "(3) raise to the power 3/2<br><BR>\n" );
document.write( "We want to perform those three operations using two functions.  We have two clear choices:<br><BR>\n" );
document.write( "(a) We can combine operations (1) and (2) into a single function:<BR>\n" );
document.write( "f(x) multiplies by 5 and subtracts 3 --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+5x-3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+5x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then g(x) raises its input to the power 3/2: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+x%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"g%28x%29+=+x%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%28x%29%29+=+%285x-3%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"g%28f%28x%29%29+=+%285x-3%29%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That is probably the most \"reasonable\" way to get h(x) as the composition of two functions.<br><BR>\n" );
document.write( "But we have another obvious possibility....<br><BR>\n" );
document.write( "(b) We can leave the first operation alone and combine (2) and (3) into a single function:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+5x ALIGN=MIDDLE  ALT=\"f%28x%29+=+5x\"   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+=+%28x-3%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"g%28x%29+=+%28x-3%29%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%28x%29%29+=+%285x-3%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"g%28f%28x%29%29+=+%285x-3%29%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And in fact there are yet other possibilities.  We could perform the single operation \"raise to the power 3/2\" as a sequence of two operations -- either \"raise to the power 3 and then take the square root\" or \"raise to the 1/2 power and then raise to the power 3\".<br><BR>\n" );
document.write( "That gives us two more ways to get h(x) as the composition of two functions:<br><BR>\n" );
document.write( "(c) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%285x-3%29%5E3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%285x-3%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+x%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"g%28x%29+=+x%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%28x%29%29+=+%28%285x-3%29%5E3%29%5E%281%2F2%29+=+%285x-3%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"g%28f%28x%29%29+=+%28%285x-3%29%5E3%29%5E%281%2F2%29+=+%285x-3%29%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "or<br><BR>\n" );
document.write( "(d) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%285x-3%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%285x-3%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+x%5E3 ALIGN=MIDDLE  ALT=\"g%28x%29+=+x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28f%28x%29%29+=+%28%285x-3%29%5E%281%2F2%29%29%5E3+=+%285x-3%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"g%28f%28x%29%29+=+%28%285x-3%29%5E%281%2F2%29%29%5E3+=+%285x-3%29%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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