Question 1183082
Two functions f and g are defined by f : x maps to x-1/x+1 , x is not equal to -1 and g : x maps to mx + c,where m and c are constants. Find an expression for f^-1. Given that g^-1(3) = f^-1(2) and that f^-1g(4) = 1,find the value of m and c
<pre>{{{system(matrix(2,3, f(x), "=", (x - 1)/(x + 1), y, "=", (x- 1)/(x + 1)), matrix(1,8, x, "=", (y - 1)/(y + 1), "------", Interchanging, x, and, y), matrix(1,5, x(y + 1), "=", y - 1, "-----", Cross-multiplying), matrix(5,3, xy + x, "=", y - 1, xy - y, "=", - x - 1, y(x - 1), "=", - x - 1, y, "=", (- x - 1)/(x - 1), highlight(highlight_green(f^(- 1)(x))), "=", highlight(highlight_green((- x - 1)/(x - 1)))), matrix(1,7, f^(- 1)(2), "=", (- 2 - 1)/(2 - 1), "=", (- 3)/1, "=", - 3)))}}}          {{{system( 
matrix(2,3, g(x), "=", mx + c, y, "=", mx + c), matrix(1,8, x, "=", my + c, "------", Interchanging, x, and, y), matrix(6,3, x - c, "=", my, (x - c)/m, "=", y, (x - c)/m, "=", g^(- 1)(x), (3 - c)/m, "=", g^(- 1)(3), g(x), "=", mx + c, g(4), "=", 4m + c))}}}  
     
                                                                                                                                                                                            
       Since {{{matrix(1,3, g^(- 1)(3), "=", f^(- 1)(2))}}}, we get:
             {{{system(matrix(1,3, (3 - c)/m, "=", - 3), 
matrix(1,5, 3 - c, "=", - 3m, "-----", Cross-multiplying), matrix(1,6, 3, "=", c - 3m, "-------", eq, "(i)"))}}}

  Since {{{matrix(1,3, f^(- 1)(x), "=", (- x - 1)/(x - 1))}}} and {{{matrix(1,3, g(4), "=", 4m + c)}}}, then {{{matrix(1,7, f^(- 1)(g(4)), "=", f^(- 1)(4m + c), "=", (- (4m + c) - 1)/(4m + c - 1), "=", (- 4m - c - 1)/(4m + c - 1))}}}    ;      {{{matrix(1,4, f^(- 1)(g(4)), also, "=", 1)}}}
We then get: {{{system(matrix(1,3, (- 4m - c - 1)/(4m + c - 1), "=", 1),
matrix(1,5, - 4m - c - 1, "=", 4m + c - 1, "-----", Cross-multiplying), matrix(2,3, - 8m, "=", 2c, (- 8m)/2, "=", c), matrix(1,6, - 4m, "=", c, "-------", eq, "(ii)"))}}}

                      3 = c - 3m ----- eq (i)
                      3 = - 4m - 3m ------- Substituting - 4m for c in eq (i)
                      3 = - 7m
                     {{{highlight_green(matrix(1,3, highlight(highlight_green(- 3/7)), "=", highlight(highlight_green(m))))}}}

                      4m = c ----- eq (ii)
                      4m = c ------- Substituting - 4m for c in eq (ii)
                      {{{highlight_green(matrix(2,3, - 4(- 3/7), "=", c, highlight(highlight_green(12/7)), "=", highlight(highlight_green(c))))}}}</pre>