find range 
f(x) = 

−x − 1    	if x < −1
0	if −1 ≤ x ≤ 1
x + 1	if x > 1


To find the range of the piecewise function:

\[
f(x) = 
\begin{cases} 
-x - 1 & \text{if } x < -1 \\
0 & \text{if } -1 \leq x \leq 1 \\
x + 1 & \text{if } x > 1
\end{cases}
\]

### Step 1: Analyze each case

1. **For \( x < -1 \):**  
   The function is \( f(x) = -x - 1 \).  
   This is a linear equation with a negative slope, meaning as \( x \) decreases (goes to negative infinity), \( f(x) \) increases to positive infinity.  
   As \( x \) approaches -1 (but remains less than -1), \( f(x) \) approaches 0:
   \[
   \lim_{x \to -1^-} f(x) = -(-1) - 1 = 0
   \]
   So, the function takes values from \( (-\infty, 0) \) in this interval.

2. **For \( -1 \leq x \leq 1 \):**  
   The function is constant \( f(x) = 0 \).  
   So, for this interval, the function only takes the value 0.

3. **For \( x > 1 \):**  
   The function is \( f(x) = x + 1 \).  
   This is a linear equation with a positive slope, meaning as \( x \) increases, \( f(x) \) also increases.  
   As \( x \) approaches 1 from the right, \( f(x) \) approaches 2:
   \[
   \lim_{x \to 1^+} f(x) = 1 + 1 = 2
   \]
   So, for \( x > 1 \), the function takes values from \( (2, \infty) \).

### Step 2: Combine the results

- For \( x < -1 \), the range is \( (-\infty, 0) \).
- For \( -1 \leq x \leq 1 \), the range is \( \{0\} \) (just 0).
- For \( x > 1 \), the range is \( (2, \infty) \).

### Step 3: Final range of the function

The range of the function \( f(x) \) is:
\[
(-\infty, 0) \cup \{0\} \cup (2, \infty)
\]
or equivalently, 
\[
(-\infty, 0] \cup (2, \infty)
\]

This represents all values less than or equal to 0 and greater than 2.

---

Would you like further details or have any questions? Here are some related questions to consider:

1. How would the graph of this function look?
2. What are the domain and continuity properties of this function?
3. What happens to the range if we modify the boundaries of the piecewise function?
4. Can the function be inverted, and if so, what would the inverse be?
5. How do we find the range of more complex piecewise functions?

**Tip:** To find the range of a piecewise function, break it down into intervals and analyze the behavior of the function in each one.

find range f(x) = -x - 1 if x < -1, 0 if -1 ≤ x ≤ 1, x + 1 if x > 1

This article explains how to find the range of a piecewise function with linear components. The function is split into three cases: f(x) = -x - 1 for x < -1, f(x) = 0 for -1 ≤ x ≤ 1, and f(x) = x + 1 for x > 1. Learn how to analyze the range of each segment and combine the results for the final range.

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