Consider the function f(x) = ⁻¹ ( 2 ⁻¹ ( a x^2 - 6x + 8 ) ) , where a > 0 . If the domain of f(x) contains exactly one isolated point (a point that is not part of any continuous interval), then the value of a + is

Step-by-step solution

For the outer ⁻¹ function to be defined, its argument must lie in [-1, 1] : -1 2 ⁻¹ ( a x^2 - 6x + 8 ) 1 - 2 ⁻¹ ( a x^2 - 6x + 8 ) 2 This condition is always satisfied for any valid input to the ⁻¹ function. Thus, we only need to find the domain of the inner function, which requires: -1 a x^2 - 6x + 8 1 Since a > 0 , this is equivalent to: |x^2 - 6x + 8| a This splits into two cases: Case 1: x^2 - 6x + 8 a x^2 - 6x + (8 - a) 0 This inequality will either be true for all real x or yield a union of continuous interva