If the function f(x) = ⁻¹ ( x-3 2 ) + 1 x^2 - 7x + 12 + _e(5-x) is defined for real values of x , then the sum of all integer values of x in the domain of f(x) is:

Step-by-step solution

The domain of f(x) is the intersection of the domains of its three terms. For ⁻¹ ( x-3 2 ) , we must have: -1 x-3 2 1 -2 x-3 2 1 x 5 For 1 x^2 - 7x + 12 , the expression inside the square root must be strictly positive: x^2 - 7x + 12 > 0 (x-3)(x-4) > 0 x (- , 3) (4, ) For _e(5-x) , the argument must be strictly positive: 5 - x > 0 x Taking the intersection of these three conditions: x [1, 5] ((- , 3) (4, )) (- , 5) x [1, 3) (4, 5) The integer values of x in this domain are 1 and 2 . Sum of integers = 1 + 2 = 3 . An