document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170516>Question 1170516</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f defined on (−∞,∞) by<BR>\n" );
document.write( "f(x)=22−58x−14(8+29x)7\r<BR>\n" );
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document.write( "is one-to-one and has range (−∞,∞).<BR>\n" );
document.write( "Find f−1(38).\r<BR>\n" );
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document.write( "Answer :<BR>\n" );
document.write( "(Do NOT USE DECIMALS if your answer is a fraction or an irrational number) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851159 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=851159\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $f(x) = 22 - 58x - 14(8+29x)^7$.<BR>\n" );
document.write( "We want to find $f^{-1}(38)$.<BR>\n" );
document.write( "Let $y = f(x)$.<BR>\n" );
document.write( "We want to find $x$ such that $f(x) = 38$.<BR>\n" );
document.write( "So we need to solve the equation:<BR>\n" );
document.write( "$38 = 22 - 58x - 14(8+29x)^7$.<BR>\n" );
document.write( "$38 - 22 = -58x - 14(8+29x)^7$.<BR>\n" );
document.write( "$16 = -58x - 14(8+29x)^7$.<BR>\n" );
document.write( "$16 + 58x = -14(8+29x)^7$.<BR>\n" );
document.write( "$-8 - 29x = 7(8+29x)^7$.\r<BR>\n" );
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document.write( "Let $u = 8+29x$. Then $-u = 7u^7$.<BR>\n" );
document.write( "$u^7 = -\frac{u}{7}$.<BR>\n" );
document.write( "$u^7 + \frac{u}{7} = 0$.<BR>\n" );
document.write( "$u(u^6 + \frac{1}{7}) = 0$.<BR>\n" );
document.write( "Since $u^6 + \frac{1}{7} > 0$ for all real $u$, we must have $u = 0$.<BR>\n" );
document.write( "$8+29x = 0$.<BR>\n" );
document.write( "$29x = -8$.<BR>\n" );
document.write( "$x = -\frac{8}{29}$.\r<BR>\n" );
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document.write( "Now, we verify this value.<BR>\n" );
document.write( "$f(-\frac{8}{29}) = 22 - 58(-\frac{8}{29}) - 14(8+29(-\frac{8}{29}))^7$.<BR>\n" );
document.write( "$= 22 + 2(8) - 14(8-8)^7$.<BR>\n" );
document.write( "$= 22 + 16 - 14(0)^7$.<BR>\n" );
document.write( "$= 38 - 0$.<BR>\n" );
document.write( "$= 38$.\r<BR>\n" );
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document.write( "Thus, $f^{-1}(38) = -\frac{8}{29}$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{-8/29}$<BR>\n" );
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