document.write( "Question 1196346: Hello, can anybody explain this problem to me? Thanks:
\n" ); document.write( "Let f be a one-to-one function and c be a real number. State an expression for
\n" ); document.write( "the inverse of g(x) = f (x + c) in terms of f^−1, and prove that your formula is correct. \n" ); document.write( "
Algebra.Com's Answer #829138 by ikleyn(52797)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Hello, can anybody explain this problem to me? Thanks:
\n" ); document.write( "Let f be a one-to-one function and c be a real number. State an expression for
\n" ); document.write( "the inverse of g(x) = f (x + c) in terms of f^−1, and prove that your formula is correct.
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document.write( "We are given two functions, f(x) and g(x) = f(x+c), which map the set of real numbers R\r\n" );
document.write( "one-to-one in R.  We express it by using this diagram with these standard symbols of mapping \r\n" );
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document.write( "                  f, g :  \"R%5BL%5D\" ---> \"R%5BR%5D\"     (1)\r\n" );
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document.write( "                          {x}       {y}\r\n" );
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document.write( "For more expressiveness, we use variable x for real numbers of the left side R = \"R%5BL%5D\"\r\n" );
document.write( "and use variable y for real numbers of the right side R = \"R%5BR%5D\", as shown in the formula (1).\r\n" );
document.write( "So we write expressions like these\r\n" );
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document.write( "                    y = f(x), y = g(x).\r\n" );
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document.write( "                   Couple of useful notes\r\n" );
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document.write( "    (1)  According to the problem's context, functions f and g are defined over entire R.\r\n" );
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document.write( "    (2)  The image of f is the same as the image of g. \r\n" );
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document.write( "    I don't want to spend time, words and place for explaining it; \r\n" );
document.write( "    it should be clear for the reader, whose level is adequate to the problem.\r\n" );
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document.write( "    +-------------------------------------------------------------------------------------------+\r\n" );
document.write( "    |    The problem asks to express \"g%5E%28-1%29%28y%29\" in terms of \"f%5E%28-1%29\".                              |\r\n" );
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document.write( "    |    Notice that this requirement does not specify, which argument of \"f%5E%28-1%29\" to use.        |\r\n" );
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document.write( "    |            This uncertainty OPENS the way to move forward.                                |\r\n" );
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document.write( "Let y belongs to the image of g  (which is the same as the image of f).\r\n" );
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document.write( "By the definition of the inverse function,  \r\n" );
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document.write( "    \"g%5E%28-1%29%28y%29\" = \"x\",      (2)\r\n" );
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document.write( "where x is the unique real number (in the left side R of (1)), such that g(x) = y.\r\n" );
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document.write( "Next, we know that  \"f%5E%28-1%29of%28x%29\" = x  (the composition of f and  \"f%5E%28-1%29\"  is the identical map).\r\n" );
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document.write( "Therefore, we can re-write formula (2) in this way\r\n" );
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document.write( "    \"g%5E%28-1%29%28y%29\" = \"f%5E%28-1%29%28f%28x%29%29\",    (3)\r\n" );
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document.write( "where x is the unique real number (in the left side R of (1)), such that g(x) = y.\r\n" );
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document.write( "        You may consider (3) as the desired expression for \"g%5E%28-1%29%28y%29\",\r\n" );
document.write( "                        or very close to it.\r\n" );
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document.write( "Or, if you want to have an analog of (3), where all the parts and all the terms of \r\n" );
document.write( "the right side are expressed in terms of f(x) only, you can replace g(x) = y by f(x-c) = y.\r\n" );
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document.write( "Then your final answer will sound this way\r\n" );
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document.write( "    \"g%5E%28-1%29%28y%29\" = \"f%5E%28-1%29%28f%28x%29%29\",      (4)\r\n" );
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document.write( "or this way\r\n" );
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document.write( "    \"g%5E%28-1%29%28y%29\" = \"f%5E%28-1%29%28g%28x-c%29%29\",    (5)\r\n" );
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document.write( "where x is the unique real number (in the left side R of (1)), such that f(x-c) = y.\r\n" );
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\n" ); document.write( "\n" ); document.write( "I understand that for beginner reader this reasoning may seem hard,
\n" ); document.write( "but you receive what you request exactly as is.\r
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\n" ); document.write( "\n" ); document.write( "To understand it in full, you need read it very attentively,
\n" ); document.write( "think on each word and each sentence and use diagrams.\r
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\n" ); document.write( "\n" ); document.write( "If you have question, do not hesitate to ask.\r
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\n" ); document.write( "\n" ); document.write( "        At which school / college / university / thinking center
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\n" ); document.write( "\n" ); document.write( "Comment from student: Thanks for helping me out! To respond to your question,
\n" ); document.write( "this is a problem from the hw for Umtymp Calc, the youth math program at U of M.\r
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\n" ); document.write( "\n" ); document.write( "My response.   I got your message as a comment.  Thanks for it.\r
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\n" ); document.write( "\n" ); document.write( "I am proud by myself,  too . . . \r
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\n" ); document.write( "\n" ); document.write( "In the  Internet,  I found the info about this program and this school,
\n" ); document.write( "and I place it here - - - for memory.\r
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\n" ); document.write( "\n" ); document.write( "                                About   UMTYMP \r
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\n" ); document.write( "\n" ); document.write( "                https://cse.umn.edu/mathcep/about-umtymp\r
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document.write( "    The University of Minnesota Talented Youth Mathematics Program (UMTYMP) \r\n" );
document.write( "    has been one of the nation's premier accelerated programs for students \r\n" );
document.write( "    who are highly talented in mathematics over the last forty years. \r\n" );
document.write( "    UMTYMP is unique in terms of the number of students, length of the program, \r\n" );
document.write( "    scope of the curriculum, and the number of University of Minnesota credits \r\n" );
document.write( "    granted to students in middle school and high school.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Come again to this forum soon to learn something new  ( ! )\r
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