If the domain of the function f(x) = ⁻¹ ( |x| - a b ) + 1 _e(c - x) , where a, b, and c are positive constants, is [-8, -2] [2, 5) - 4 , then the value of 2a + 3b + c is equal to :

Step-by-step solution

For the function f(x) to be defined, the following conditions must be satisfied: First, the argument of the logarithm must be positive: c - x > 0 x Second, the denominator cannot be zero, which means the argument of the logarithm cannot be 1 : _e(c - x) 0 c - x 1 x c - 1 Third, the argument of the inverse cosine function must lie in [-1, 1] : -1 |x| - a b 1 Since b > 0 , this simplifies to: -b |x| - a b a - b |x| a + b This inequality implies that x lies in the union of two intervals: x [-(a + b), -(a - b)] [a - b,