-1 if z≤-1 Sketch a graph of f(x) = 2x+1 if-1<x<1 2 if 1

The graph above represents the piecewise function:

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

### Key points:
- For \( x \leq -1 \), the function is a constant horizontal line at \( y = -1 \).
- For \( -1 < x < 1 \), the function is a line with a slope of 2.
- For \( x \geq 1 \), the function is constant at \( y = 2 \).

Do you want more details or have any questions?

Here are some related questions:
1. How do we identify discontinuities in a piecewise function?
2. What are the key features of linear functions like \( 2x + 1 \)?
3. How do we find where a piecewise function is continuous?
4. Can this function be differentiated everywhere? Why or why not?
5. How would the graph change if \( f(x) = 3x - 2 \) for \( -1 < x < 1 \)?

**Tip:** For piecewise functions, check for continuity and differentiability at the transition points between different segments.

The query submitted by the user: -1 if z ≤ -1. Sketch a graph of f(x) = 2x + 1 if -1 < x < 1, 2 if 1

Learn how to graph the piecewise function f(x), which combines constant and linear segments. The function includes three parts: a constant at y = -1, a linear section 2x + 1 for -1 < x < 1, and a constant at y = 2 for x ≥ 1. This problem is suitable for high school students in Grades 9-12, covering key concepts in graphing piecewise functions and analyzing continuity.

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