Let f(x) = x^2 - ax + b |x| - c + _e(d - x) , where a, b, c, d are real constants and c > 0 . If the domain of f(x) is (- , 1] (4, 8) -4 , then the value of a + b + c + d is:

Step-by-step solution

The domain of f(x) is determined by three conditions: 1) For _e(d - x) , we require d - x > 0 x 2) The denominator cannot be zero, so |x| - c 0 x c . The given domain excludes -4 and has an open boundary at 4 . Since c > 0 , we must have c = 4 . This excludes both x = 4 and x = -4 . 3) For the square root, we require x^2 - ax + b 0 . The critical points where the domain changes from included to excluded (before applying the denominator and logarithm constraints) are 1 and 4 . Thus, the roots of the equation x^2 - a