https://file.aiquickdraw.com/mathbot/file/1726104718466q9irhrfg.jpg

The image shows a piecewise function \( y = f(x) \) defined as:

\[
f(x) = 
\begin{cases} 
x, & \text{if } x \in (-\infty, 1) \\
x^2, & \text{if } x \in [1, 4] \\
2x, & \text{if } x \in (4, \infty)
\end{cases}
\]

This is a function defined over different intervals of \(x\):

1. For \( x \in (-\infty, 1) \), \( f(x) = x \).
2. For \( x \in [1, 4] \), \( f(x) = x^2 \).
3. For \( x \in (4, \infty) \), \( f(x) = 2x \).

Would you like a deeper explanation of how to graph this or explore properties of this function (e.g., continuity, differentiability)?

Here are some related questions to deepen your understanding:
1. What are the points of continuity or discontinuity in this piecewise function?
2. How would you differentiate this function for each interval?
3. What is the slope of the function at different points?
4. How would the graph of this function look across the entire domain?
5. How does the function behave as \( x \to -\infty \) and \( x \to +\infty \)?

**Tip:** When graphing piecewise functions, make sure to plot each part separately, paying attention to open and closed intervals.

The original query submitted by the user involves a piecewise function with different expressions for different intervals of x.

This problem explores a piecewise function defined by different expressions for different intervals: a linear expression for x in (-∞, 1), a quadratic function for x in [1, 4], and a linear function for x in (4, ∞). Explore how to solve and graph piecewise functions, ensuring continuity and differentiability across intervals.

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