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 piecewise function and its components. The function is defined as: y = -x - 2 for -2 ≤ x ≤ -1, y = x for -1 < x ≤ 1, and y = -x + 2 for 1 < x ≤ 2.
2. Step 2: Determine the domain of the function. The domain is the set of all x-values for which the function is defined. Here, the domain is the union of the intervals: [-2, -1], (-1, 1], and (1, 2].
3. Step 3: Analyze each piece of the function to find the range. For y = -x - 2, as x goes from -2 to -1, calculate the corresponding y-values. For y = x, as x goes from -1 to 1, calculate the y-values. For y = -x + 2, as x goes from 1 to 2, calculate the y-values.
4. Step 4: Combine the ranges from each piece to find the overall range of the function. Consider the y-values obtained from each interval and ensure there are no gaps.
5. Step 5: Verify the continuity and endpoints of the function. Check the values at the boundaries of each interval to ensure they are included in the range, and confirm that the function transitions smoothly between pieces where applicable.

Key Concepts

Domain

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In this case, the function is piecewise defined, meaning it has different expressions for different intervals of x. To find the domain, we need to identify the intervals specified in the piecewise function and combine them to determine the overall set of x-values.

Range

The range of a function is the set of all possible output values (y-values) that the function can produce based on its domain. For piecewise functions, we must evaluate each piece separately to find the corresponding y-values and then combine these results to determine the overall range. This often involves calculating the function's values at the endpoints of the intervals and any critical points within them.

Piecewise Function

A piecewise function is defined by different expressions over different intervals of its domain. Each piece of the function applies to a specific range of x-values, and understanding how to analyze each segment is crucial for determining the overall behavior of the function. In this question, the function is defined in three segments, each with its own formula and domain restrictions.