Determine the domain for the first term, g(x) = ^-1((2-|x|)/(4)).

Step 1: Determine the domain for the first term, g(x) = ^-1((2-|x|)/(4)). For ^-1(u) to be defined, the argument u must satisfy -1 u 1. So, we must have: -1 (2-|x|)/(4) 1 Multiply all parts by 4: -4 2-|x| 4 Subtract 2 from all parts: -4-2 -|x| 4-2 -6 -|x| 2 Multiply all parts by -1 and reverse the inequality signs: -2 |x| 6 Since |x| is always non-negative, the condition |x| -2 is always true. Thus, we only need to consider |x| 6. This implies -6 x 6. So, the domain for the first term is D_1 = [-6, 6]. Step 2: Determine the domain for the second term, h(x) = \_e(3-x)\^-1 = (1)/(_e(3-x)). For this term to be defined, two conditions must be met: 1. The argument of the logarithm must be positive: 3-x > 0 x < 3 2. The denominator cannot be zero, meaning _e(3-x) 0. This implies 3-x e^0. 3-x 1 x 2 So, the domain for the second term is D_2 = (-, 3) \2\. Step 3: Find the overall domain of f(x). The domain of f(x) is the intersection of D_1 and D_2: D = D_1 D_2 = [-6, 6] ((-, 3) \2\) The intersection of [-6, 6] and (-, 3) is [-6, 3). Then, we must exclude x=2 from this interval. So, the domain is D = [-6, 3) \2\, which can be written as [-6, 3] - \2\. Step 4: Compare the derived domain with the given form [-, ] - \\. By comparing D = [-6, 3] - \2\ with [-, ] - \\, we can identify the values: = -6 = 6 = 3 = 2 Step 5: Calculate + + . + + = 6 + 3 + 2 = 11 The final answer is 11. Send me the next one 📸