 the updated piecewise function looks like this:

f(x)={−1,if x≤−12x+1,if −1<x≤12,if x>1*f*(*x*)=⎩⎨⎧​−1,2*x*+1,2,​if *x*≤−1if −1<*x*≤1if *x*>1​

The piecewise function you provided can be clarified as follows:

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

### Explanation:

- **For \( x \leq -1 \)**: The value of the function is \( -1 \).
- **For \( -1 < x \leq 1 \)**: The function is linear, with the expression \( f(x) = 2x + 1 \).
- **For \( x > 1 \)**: The function takes a constant value of 2.

Would you like further details or visual representation of this function?

### Related Questions:
1. What is the domain of this piecewise function?
2. How would you find the limits of \( f(x) \) as \( x \to -1 \) and \( x \to 1 \)?
3. What is the range of the function?
4. How does the function behave around the discontinuities at \( x = -1 \) and \( x = 1 \)?
5. How would you graph this piecewise function?

**Tip**: For piecewise functions, always check continuity and differentiability at the points where the function changes definition!

The updated piecewise function looks like this: f(x) = {−1, if x ≤ −1; 2x + 1, if −1 < x ≤ 1; 2, if x > 1}

Learn how to solve and graph the piecewise function f(x) with segments defined by inequalities: f(x) = −1 for x ≤ −1, f(x) = 2x + 1 for −1 < x ≤ 1, and f(x) = 2 for x > 1. Explore key concepts like continuity, linear functions, and boundary behavior. Suitable for students in Grades 10-12.

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