document.write( "Question 1199428: Find f(x) if f(g(x))= (f(x))(x+1)(x+2)-1 and g(x) = x^3 +2x^2 -x -2 \n" ); document.write( "

Algebra.Com's Answer #833317 by ikleyn(53765) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Find f(x) if f(g(x))= (f(x))(x+1)(x+2)-1 and g(x) = x^3 +2x^2 -x -2
\n" ); document.write( "~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            The problem's formulation looks strange and, probably (from my point of view), is incomplete.\r
\n" ); document.write( "\n" ); document.write( "            To be complete, the problem must say that f(x) is a polynomial function.\r
\n" ); document.write( "\n" ); document.write( "            But I will assume it \" by default \" or \" from the context \".\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Then from the degree analysis, f(x) must be a linear function (a binomial)\r\n" );
document.write( "\r\n" );
document.write( "    f(x) = ax + b,\r\n" );
document.write( "\r\n" );
document.write( "and my task is to find coefficients \"a\" and \"b\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For it, it is enough to know f(x) in two points.\r\n" );
document.write( "\r\n" );
document.write( "I will choose the points x= -1 and x= 0, and will coplete the solution step by step.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(a)  if x = -1, then  g(-1) = (-1)^3 + 2*(-1)^2 - (-2) - 2 = -1 + 2 + 1 - 2 = 0.\r\n" );
document.write( "\r\n" );
document.write( "     Therefore, the given equality  f(g(x))= (f(x))(x+1)(x+2)-1  takes the form\r\n" );
document.write( "\r\n" );
document.write( "         f(0) = (f(-1))*(-1 +1 )*(-1 + 2) - 1 = (f(-1))*0*1 - 1 = -1.   (*)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(b)  if x = 0, then g(0) = 0^3 + 2*0^2 - 0 - 2 = -2.\r\n" );
document.write( "\r\n" );
document.write( "     Therefore, the given equality  f(g(x))= (f(x))(x+1)(x+2)-1  takes the form\r\n" );
document.write( "\r\n" );
document.write( "         f(-2) = (f(0))*(0 +1 )*(0 + 2) - 1 = (f(0))*1*2 - 1 = 2*f(0) - 1.\r\n" );
document.write( "\r\n" );
document.write( "                 Substitute here  f(0) = -1  from (*),  and you will get \r\n" );
document.write( "\r\n" );
document.write( "                     f(-2) = 2*(-1) - 1 = -2 - 1 = -3. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(c)  Thus we have these two equations to find coefficients \"a\" and \"b\" in  f(x) = ax + b\r\n" );
document.write( "\r\n" );
document.write( "         f(0)  = a*0    + b = -1,     (1)\r\n" );
document.write( "\r\n" );
document.write( "         f(-2) = a*(-2) + b = -3.     (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "      Equation (1) gives b = -1.\r\n" );
document.write( "\r\n" );
document.write( "      Then equation (2) gives \r\n" );
document.write( "\r\n" );
document.write( "                 -2a - 1 = -3  --->  -2a = -3 + 1 = -2  --->  a = 1.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus f(x) = x-1.      ANSWER\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "CHECK. Then left  side  f(g(x))  is  (x^3 +2x^2 -x -2) - 1 = x^3 + 2x^2 - x -3.\r\n" );
document.write( "\r\n" );
document.write( "            Right side           is  (x-1)*(x+1)*(x+2) - 1 = (x-1)*(x^2 + 3x + 2) - 1 = \r\n" );
document.write( "\r\n" );
document.write( "                                     = x^3 + 3x^2 + 2x - x^2 - 3x - 2 - 1 = x^3 + 2x^2 - x -3.\r\n" );
document.write( "\r\n" );
document.write( "       Both sides are the same (are identical).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus the answer is verified/confirmed.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );