document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209610>Question 1209610</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose f is a function such that f(f(x)) = x for all x in the domain of f, and suppose that f(-3) = 0.  If f(x) = ax + b, then find a and b.  Find all solutions to f(x) = 5.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849671 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849671\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
document.write( "\n" );
document.write( "1. **Use the given information about f(x):**\r<BR>\n" );
document.write( "\n" );
document.write( "We know f(x) = ax + b and f(f(x)) = x.  Substituting the expression for f(x) into the composite function:\r<BR>\n" );
document.write( "\n" );
document.write( "f(f(x)) = f(ax + b) = a(ax + b) + b = a²x + ab + b = x\r<BR>\n" );
document.write( "\n" );
document.write( "2. **Solve for a and b:**\r<BR>\n" );
document.write( "\n" );
document.write( "For the equation a²x + ab + b = x to hold for all x, the coefficients of x must be equal, and the constant terms must be equal.  This gives us the following system of equations:\r<BR>\n" );
document.write( "\n" );
document.write( "* a² = 1<BR>\n" );
document.write( "* ab + b = 0\r<BR>\n" );
document.write( "\n" );
document.write( "From a² = 1, we have two possibilities: a = 1 or a = -1.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Case 1: a = 1**<BR>\n" );
document.write( "   Substituting into ab + b = 0, we get b + b = 0, so 2b = 0, and b = 0.<BR>\n" );
document.write( "   Thus, f(x) = x.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Case 2: a = -1**<BR>\n" );
document.write( "   Substituting into ab + b = 0, we get -b + b = 0. This equation is true for any value of b.\r<BR>\n" );
document.write( "\n" );
document.write( "3. **Use f(-3) = 0 to find b (in the case a=-1):**\r<BR>\n" );
document.write( "\n" );
document.write( "We are given that f(-3) = 0.  Using f(x) = ax + b:\r<BR>\n" );
document.write( "\n" );
document.write( "0 = a(-3) + b\r<BR>\n" );
document.write( "\n" );
document.write( "If a = -1, then 0 = -(-3) + b, so 0 = 3 + b, and b = -3.\r<BR>\n" );
document.write( "\n" );
document.write( "Thus, if a = -1, then b = -3, and f(x) = -x - 3.\r<BR>\n" );
document.write( "\n" );
document.write( "4. **Determine which solution is correct:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **If f(x) = x:** Then f(-3) = -3, which contradicts the given information f(-3) = 0.  So, f(x) = x is not a solution.\r<BR>\n" );
document.write( "\n" );
document.write( "* **If f(x) = -x - 3:** Then f(-3) = -(-3) - 3 = 3 - 3 = 0, which matches the given information.  Also, f(f(x)) = f(-x - 3) = -(-x - 3) - 3 = x + 3 - 3 = x.  This solution works.\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, a = -1 and b = -3, and f(x) = -x - 3.\r<BR>\n" );
document.write( "\n" );
document.write( "5. **Solve f(x) = 5:**\r<BR>\n" );
document.write( "\n" );
document.write( "We want to find x such that f(x) = 5.  Using f(x) = -x - 3:\r<BR>\n" );
document.write( "\n" );
document.write( "5 = -x - 3<BR>\n" );
document.write( "8 = -x<BR>\n" );
document.write( "x = -8\r<BR>\n" );
document.write( "\n" );
document.write( "So, the solution to f(x) = 5 is x = -8.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );