document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1157859>Question 1157859</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What happens if we graph both  f  and  f^{-1}  on the same set of axes, using the x-axis for the input to both  f  and  f^{-1}  ? \r<BR>\n" );
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document.write( "[Suggestion: go to www.desmos.com/calculator and type  y=x^3  {-2 < x < 2},  y=x^{1/3}  {  -  2 < x < 2}, and y = x {  -  2 < x < 2}, and describe the relationship between the three curves.] Then post your own example discussing the difficulty of graph both  f  and  f^{-1}  on the same set of axes. \r<BR>\n" );
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document.write( "Suppose  f:R \rightarrow R   is a function from the set of real numbers to the same set with  f(x)=x+1 . We write  f^{2}  to represent  f \circ f  and  f^{n+1}=f^n \circ f . Is it true that  f^2 \circ f = f \circ f^2 ? Why? Is the set { g:R \rightarrow R   l  g \circ f=f \circ g } infinite? Why?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #780750 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=780750\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> here's a reference on the properties of functions that are inverse of each other.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.analyzemath.com/inversefunction/properties_inverse.html\" target = \"_blank\">https://www.analyzemath.com/inversefunction/properties_inverse.html</a><BR>\n" );
document.write( "the graph of inverses are reflective of each other across the line y = x.<BR>\n" );
document.write( "this can be shown in the following graph.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2020/050201.jpg\" alt=\"$$$\" ><BR>\n" );
document.write( "the red line is the graph of the original equation of y = x^3.<BR>\n" );
document.write( "the blue line is the graph of the inverse equation of y = x^(1/3)<BR>\n" );
document.write( "the lin y = -x + 4 is there to allow me to show you that the point (x,y) on the graph of the original equation is opposite and equidistant from the line y = x and the reflective point is (y,x) on the graph of the inverse equation.<BR>\n" );
document.write( "specifically, the point (1.379,2.621) on the graph of the original equation is equidistant from the line y = x as the point (2.621,1.379) on the inverse equation.\r<BR>\n" );
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document.write( "i'm not sure i understand the last part of your question, so i won't try to answer it.<BR>\n" );
document.write( "hopefully, what i have provided is helpful to you.<BR>\n" );
document.write( "most of the properties of inverse functions are in the reference.\r<BR>\n" );
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document.write( "there is another properties of inverse functions that may be what you are alluding to in the last part of your question.<BR>\n" );
document.write( "that is that (fog)x) = x if f and g are inverse functions.<BR>\n" );
document.write( "in your original example, that is confiemd as shown below.<BR>\n" );
document.write( "(fog)(x) = f(g(x)).<BR>\n" );
document.write( "(gof(x) = g(f(x)).<BR>\n" );
document.write( "if they are inverses of each other, the f(g(x)) = x and g(f(x)) = x.<BR>\n" );
document.write( "specifically.<BR>\n" );
document.write( "f(g(x)) = (x^(1/3))^3 = x<BR>\n" );
document.write( "g(f(x)) = (x^3)^(1/3) = x<BR>\n" );
document.write( "check the reference.<BR>\n" );
document.write( "lots of good stuff in there.\r<BR>\n" );
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document.write( "  </SPAN>\n" );
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