Let S be the set of all integers x in the interval [1, 10] that satisfy the equation ⁻¹( x) + ⁻¹( x) = . The sum of all elements of S is equal to

Step-by-step solution

Let f(x) = ⁻¹( x) + ⁻¹( x) . We analyze f(x) in intervals of length /2 : For x [0, /2] , f(x) = x + x = 2x . For x [ /2, ] , f(x) = ( - x) + x = . For x [ , 3 /2] , f(x) = ( - x) + (2 - x) = 3 - 2x . For x [3 /2, 2 ] , f(x) = (x - 2 ) + (2 - x) = 0 . For x [2 , 5 /2] , f(x) = (x - 2 ) + (x - 2 ) = 2x - 4 . For x [5 /2, 3 ] , f(x) = (3 - x) + (x - 2 ) = . In general, f(x) = for x [2n + /2, 2n + ] where n Z . For n = 0 , the interval is [ /2, ] [1.57, 3.14] . The integers in this interval are 2 and 3 . For n = 1 , th