document.write( "Question 1035668: A1. Let f be a one-to-one function whose inverse function is given by the formula\r
\n" ); document.write( "\n" ); document.write( "\"f%5E%28-1%29%28x%29=x%5E5%2B2x%5E3%2B3x%2B1\"\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "(a) Compute \"f%5E-1%281%29\" and \"f%281%29\"
\n" ); document.write( "(b) Compute the value of \"x%5B0%5D\" such that \"f%28x%5B0%5D%29+=+1\".
\n" ); document.write( "(c) Compute the value of \"y%5B0%5D\" such that \"f%5E-1%28y%5B0%5D%29=1\".\r
\n" ); document.write( "\n" ); document.write( "Thankyou for looking over! Steps and instructions will be appreciate. \n" ); document.write( "
Algebra.Com's Answer #650341 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "First I'll teach you a little bit about inverse functions.\r\n" );
document.write( "Stuff you should know before you try to do inverse problems:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This is the graph of the inverse of f(x), which we call f-1(x). \r\n" );
document.write( "\r\n" );
document.write( "The inverse of a function is the graph made by interchanging\r\n" );
document.write( "the x-coordinates with the y-coordinates of the points on\r\n" );
document.write( "the graph.\r\n" );
document.write( "\r\n" );
document.write( "That is to say that if (a,b) is a point on f(x), then\r\n" );
document.write( "(b,a) is a point on f-1(x).\r\n" );
document.write( "\r\n" );
document.write( "If the equation for f-1(x) were simpler we could\r\n" );
document.write( "calculate it and graph it.  But we can't do that here because\r\n" );
document.write( "we don't know how to solve a 5th degree polynomial for x.\r\n" );
document.write( "\r\n" );
document.write( "I will find a few points on f(x), by finding\r\n" );
document.write( "a few points on f-1(x) and reversing the coordinates.\r\n" );
document.write( "\r\n" );
document.write( "   x   f-1(x)    point on f-1(x)   point on f(x)\r\n" );
document.write( "------------------------------------------------\r\n" );
document.write( "  -1   -5.00       (-1,-5.00)       (-5.00,1)\r\n" );
document.write( "-0.9   -3.75       (-0.9,-3.75)     (-3.75,-0.9)     \r\n" );
document.write( "-0.8   -2.75       (-0.8,-2.75)     (-2.75,-0.8)\r\n" );
document.write( "-0.7   -1.95       (-0.7,-1.95)     (-1.95,-0.7)\r\n" );
document.write( "-0.5   -0.78       (-0.5,-0.78)     (-0.78,-0.5)\r\n" );
document.write( " 0.0    1.00       (0,1.00)         (1.00,0)\r\n" );
document.write( " 0.3    1.96       (0.3,1.96)       (1.96,0.3)\r\n" );
document.write( " 0.5    2.78       (0.5,2.78)       (2.78,0.5)\r\n" );
document.write( " 0.7    3.95       (0.7,3.95)       (3.95,0.7)\r\n" );
document.write( " 0.8    4.75       (0.8,4.75)       (4.75,0.8)\r\n" );
document.write( "\r\n" );
document.write( "Here are those 10 points of f(x) plotted:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we'll look at your questions:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(a) Compute f-1(1) and f(1)\r\n" );
document.write( "\r\n" );
document.write( "To find f-1(1), substitute x=1 in\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So f-1(1) = 7\r\n" );
document.write( "\r\n" );
document.write( "That means the point (1,7) is on the graph of f-1(x). \r\n" );
document.write( "\r\n" );
document.write( "We want to find f(1).\r\n" );
document.write( "\r\n" );
document.write( "We notice that since  ends with a 1.\r\n" );
document.write( "Then if x=0 then \r\n" );
document.write( "\r\n" );
document.write( "So the point (0,1) is on the graph of f-1(x),\r\n" );
document.write( "so the point (1,0) is on the graph of f(x), so\r\n" );
document.write( "\r\n" );
document.write( "f(1) = 0 \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(b) Compute the value of \"x%5B0%5D\" such that \"f%28x%5B0%5D%29+=+1\".\r\n" );
document.write( "\r\n" );
document.write( "That means (x0,1) must be on f(x), so its reverse,\r\n" );
document.write( "(1,x0) must be on f-1(x), and since f-1(1) = 7, \r\n" );
document.write( "(1,7) is on f-1(x) and that means (7,1) is on f(x), so\r\n" );
document.write( "therefore f(7)=1 so x0 = 7\r\n" );
document.write( "\r\n" );
document.write( " (c) Compute the value of \"y%5B0%5D\" such that \"f%5E-1%28y%5B0%5D%29=1\".\r\n" );
document.write( "\r\n" );
document.write( "That means (y0,1) must be on f-1(x), so its reverse,\r\n" );
document.write( "(1,y0) must be on f(x), and since f(1) = 0, \r\n" );
document.write( "(1,0) is on f(x) and that means (0,1) is on f-1(x), so\r\n" );
document.write( "therefore f-1(0)=1 so y0 = 0.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );