=⎧⎩⎨⎪⎪1−𝑥,2,𝑥+1, if 𝑥≤−1, if −1<𝑥<1, if 𝑥≥1.
 
a. 𝑓(−2)=
 
2
functionsequation editor Help on numbers.

b. 𝑓(1)=
 
functionsequation editor Help on numbers.

c. For what values of 𝑥
 is 𝑓(𝑥)=2
?

functionsequation editor Help on intervals.

d. Find the domain and range of 𝑓
. (You may find it helpful to graph this function on your own paper to find the domain and range.)

Domain: 
functionsequation editor Help on intervals.
Range:   
functionsequation editor Help on intervals.

The given function can be interpreted as a piecewise function, which is written as:

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

Let's break down and solve each part of the question.

---

**a. \( f(-2) \)**

To find \( f(-2) \), we need to check which piece of the function applies for \( x = -2 \). Since \( -2 \leq -1 \), we use the first part of the function \( f(x) = 1 - x \).

\[
f(-2) = 1 - (-2) = 1 + 2 = 3
\]

So, \( f(-2) = 3 \).

---

**b. \( f(1) \)**

Now, let's find \( f(1) \). Since \( 1 \geq 1 \), we use the third part of the function, \( f(x) = x + 1 \).

\[
f(1) = 1 + 1 = 2
\]

So, \( f(1) = 2 \).

---

**c. For what values of \( x \) is \( f(x) = 2 \)?**

We need to solve \( f(x) = 2 \) for each piece of the function.

1. For the first piece, \( f(x) = 1 - x \), we solve \( 1 - x = 2 \):
   \[
   1 - x = 2 \implies -x = 1 \implies x = -1
   \]
   So, \( x = -1 \).

2. For the second piece, \( f(x) = 2 \) is already given, so for \( -1 < x < 1 \), \( f(x) = 2 \) for all \( x \) in this interval.

3. For the third piece, \( f(x) = x + 1 \), we solve \( x + 1 = 2 \):
   \[
   x + 1 = 2 \implies x = 1
   \]
   So, \( x = 1 \).

Therefore, \( f(x) = 2 \) for \( x \in (-1, 1] \).

---

**d. Find the domain and range of \( f \)**

- **Domain**: The function is defined for all real numbers \( x \). Therefore, the domain is:
  \[
  \text{Domain} = (-\infty, \infty)
  \]

- **Range**: The function outputs different values depending on the intervals:
  - For \( x \leq -1 \), \( f(x) = 1 - x \), which is a line with decreasing values.
  - For \( -1 < x < 1 \), \( f(x) = 2 \).
  - For \( x \geq 1 \), \( f(x) = x + 1 \), which increases without bound.

  Therefore, the range is:
  \[
  \text{Range} = [2, \infty)
  \]

---

Would you like further details or clarifications?

---

**5 Related Questions:**

1. What is the behavior of \( f(x) \) as \( x \to -\infty \)?
2. How would the graph of this piecewise function look?
3. For what values of \( x \) is \( f(x) = 3 \)?
4. How does the range change if one of the intervals in the piecewise function is modified?
5. What is the significance of continuity in this piecewise function?

**Tip**: When working with piecewise functions, always check which condition applies to the specific \( x \)-value before solving!

The query submitted by the user:⎧⎩⎨⎪⎪1−𝑥,2,𝑥+1, if 𝑥≤−1, if −1<𝑥<1, if 𝑥≥1.

a. 𝑓(−2)=

b. 𝑓(1)=

c. For what values of 𝑥 is 𝑓(𝑥)=2?

d. Find the domain and range of 𝑓.

Explore a detailed breakdown of a piecewise function involving three distinct cases. This problem addresses key concepts like domain, range, and solving for specific values of f(x). Suitable for high school students in Grades 9-12.

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