document.write( "Question 1161665: Are there any functions defined on real numbers or rational numbers other than zero function and identity function in such way that f^{3}g(x) = fg(fg(x)) i.e.
\n" ); document.write( "f(f(f(g(x)))) = f(g(f(g(x)))) for all x belongs to R or Q.
\n" ); document.write( "[Note: Here f(g(x)) is composition of functions or composite function rule] \n" ); document.write( "
Algebra.Com's Answer #785275 by ikleyn(52798)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "You may take, for example, g(x) = ax + b any linear function with a =/= 0,\r
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\n" ); document.write( "\n" ); document.write( "and take f(x) = g(x) as the same function.\r
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\n" ); document.write( "\n" ); document.write( "Then you will have the desired identity.\r
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\n" ); document.write( "\n" ); document.write( "More generally, you can take g(x) as any monotonic one-to-one function  g: R ---> R  defined over all real numbers;\r
\n" ); document.write( "\n" ); document.write( "for example,   g(x) = x^3;  or   g(x) = x^5;  or   g(x) = x^7,   and so on . . . \r
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\n" ); document.write( "\n" ); document.write( "and take  f(x) = g(x).\r
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\n" ); document.write( "\n" ); document.write( "Then,  again,  you will have the desired identity.\r
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