In Exercises 35 and 36, find the (a) domain and (b) range.


𝔂 = { -x - 2, -2 ≤ x ≤ - 1
{ x, -1 < x ≤ 1
{ -x + 2, 1 < x ≤ 2

Verified step by step guidance

1. Step 1: Identify the domain of the piecewise function. The domain is the set of all possible input values (x-values) for which the function is defined. For this piecewise function, the domain is given by the intervals: -2 ≤ x ≤ -1, -1 < x ≤ 1, and 1 < x ≤ 2. Therefore, the domain is the union of these intervals: [-2, 2].
2. Step 2: Determine the range of the first piece of the function, y = -x - 2, for the interval -2 ≤ x ≤ -1. Calculate the values of y at the endpoints of the interval: when x = -2, y = -(-2) - 2 = 0; when x = -1, y = -(-1) - 2 = -1. Thus, the range for this piece is [-1, 0].
3. Step 3: Determine the range of the second piece of the function, y = x, for the interval -1 < x ≤ 1. Calculate the values of y at the endpoints of the interval: when x approaches -1 from the right, y approaches -1; when x = 1, y = 1. Thus, the range for this piece is (-1, 1].
4. Step 4: Determine the range of the third piece of the function, y = -x + 2, for the interval 1 < x ≤ 2. Calculate the values of y at the endpoints of the interval: when x approaches 1 from the right, y approaches 1; when x = 2, y = -2 + 2 = 0. Thus, the range for this piece is (0, 1].
5. Step 5: Combine the ranges from each piece of the function to find the overall range. The ranges from the three pieces are [-1, 0], (-1, 1], and (0, 1]. The overall range is the union of these intervals, which is [-1, 1].