https://file.aiquickdraw.com/mathbot/file/1728569092746201ixake.jpg

The piecewise function given in the image is defined as:

\[
f(x) = \begin{cases} 
5 & \text{if } x \leq -3 \\
|x| & \text{if } -3 < x \leq 1 \\
x^2 - 2 & \text{if } x > 1 
\end{cases}
\]

### Explanation of the function:
1. **For \( x \leq -3 \):** The function is constant and equal to 5.
2. **For \( -3 < x \leq 1 \):** The function is \( |x| \), the absolute value of \( x \), which is piecewise itself:
   - \( |x| = x \) for \( x \geq 0 \)
   - \( |x| = -x \) for \( x < 0 \)
3. **For \( x > 1 \):** The function is quadratic, \( f(x) = x^2 - 2 \), which is a parabola shifted downward by 2 units.

### Graphing Insight:
- **At \( x = -3 \):** The function jumps from the constant value \( f(x) = 5 \) to \( f(x) = |x| = 3 \).
- **From \( x = -3 \) to \( x = 1 \):** The function follows the absolute value graph, which is V-shaped and symmetric around the y-axis.
- **At \( x = 1 \):** The function moves to the quadratic part, transitioning smoothly into a parabola starting from \( f(1) = 1^2 - 2 = -1 \).

Would you like me to assist with graphing or further analysis?

### Additional Questions:
1. How would you graph the absolute value portion between \( -3 \) and \( 1 \)?
2. What are the critical points for this piecewise function?
3. How would you compute the limits at the transition points \( x = -3 \) and \( x = 1 \)?
4. Can you describe the behavior of the quadratic part \( x^2 - 2 \) as \( x \to \infty \)?
5. How would the function change if the middle part were \( x^2 \) instead of \( |x| \)?

### Tip:
When graphing piecewise functions, always carefully consider the transition points where the function changes form!

Given a piecewise function f(x) = {5, if x <= -3; |x|, if -3 < x <= 1; x^2 - 2, if x > 1}, how do you analyze and graph it?

Explore how to analyze and graph a piecewise function involving constant values, absolute value, and quadratic functions. Understand the transitions between different segments, including the absolute value function between -3 and 1, and a parabola for x > 1. Suitable for high school students in grades 9-12.

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