document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189760>Question 1189760</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For each pair of functions f and g below, find f (g (x)) and g (f(x))<BR>\n" );
document.write( "Then, determine whether f and g are inverses of each other.<BR>\n" );
document.write( "Simplify your answers as much as possible.<BR>\n" );
document.write( "(Assume that your expressions are defined for all x in the domain of the composition.<BR>\n" );
document.write( "You do not have to indicate the domain.)<BR>\n" );
document.write( "(a) f(x) =1/4x , x ≠ 0<BR>\n" );
document.write( "g(x) = 1/4x , x ≠ 0\r<BR>\n" );
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document.write( "f(g(x))= <BR>\n" );
document.write( "g(f(x))= \r<BR>\n" );
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document.write( "are f and g inverses of each other? yes or no?\r<BR>\n" );
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document.write( "(b) f(x)=x+6<BR>\n" );
document.write( "g(x)=x+6\r<BR>\n" );
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document.write( "f(g(x))= <BR>\n" );
document.write( "g(f(x))= \r<BR>\n" );
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document.write( "are f and g inverses of each other? yes or no? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #821215 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=821215\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if f(x) is the inverse of g(x), then f(g(x)) would be equal to x.<BR>\n" );
document.write( "if g(x) is the inverse of f(x), then g(f(x)) would be equal to x.\r<BR>\n" );
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document.write( "(a) f(x) = 1/4 * x , x ≠ 0<BR>\n" );
document.write( "g(x) = 1/4 * x , x ≠ 0<BR>\n" );
document.write( "f(g(x))= 1/4 * (1/4 * x) = 1/16 * x which is not equal to x.<BR>\n" );
document.write( "g(f(x))= 1/4 * (1/4 * x) = 1/16 * x which is not equal to x.<BR>\n" );
document.write( "are f and g inverses of each other? yes or no?<BR>\n" );
document.write( "answer is no.<BR>\n" );
document.write( "in fact, these equations are identical to each other.\r<BR>\n" );
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document.write( "(b) f(x)=x+6<BR>\n" );
document.write( "g(x)=x+6<BR>\n" );
document.write( "f(g(x))= (x + 6) + 6 = x + 12 which is not equal to x.<BR>\n" );
document.write( "g(f(x))= (X + 6) + 6 = X + 12 which is not equal to x.<BR>\n" );
document.write( "answer is no.<BR>\n" );
document.write( "in fact, these equations are identical to each other.\r<BR>\n" );
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document.write( "you will notice that f(x) and g(x) are the same equation, i.e. they are identical.<BR>\n" );
document.write( "as such, they can't possibly be inverses of each other.\r<BR>\n" );
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document.write( "here's a good reference on inverse functions.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut32b_inverfun.htm\" target = \"_blank\">https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut32b_inverfun.htm</a>\r<BR>\n" );
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document.write( "note that, if your equation doesn't pass the horizontal line test, then it is not a function.<BR>\n" );
document.write( "in that case, however, you can turn it into a function by restricting the domain.<BR>\n" );
document.write( "for example, in example 4 of the tutorial, if you restricted the domain to all values of x >= 0, then the equation would be a function because it would then pass the horizontal line test .\r<BR>\n" );
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document.write( "to find the inverse function of f(x0 = x + 6, you would do the following:<BR>\n" );
document.write( "let y = f(x) to get y = x + 6.<BR>\n" );
document.write( "replace y with x and x with y to get x = y + 6.<BR>\n" );
document.write( "solve for y to get y = x - 6.<BR>\n" );
document.write( "f(x) = x + 6 is the original functiojn.<BR>\n" );
document.write( "g(x) = x - 6 is the inverse fucntion.<BR>\n" );
document.write( "f(g(x)) = (x - 6) + 6 which is equal to x.<BR>\n" );
document.write( "this passes the test.<BR>\n" );
document.write( " </SPAN>\n" );
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