https://file.aiquickdraw.com/mathbot/file/1727099582563on8vpzpy.jpg

The piecewise function \( f(x) \) given in the image is:

\[
f(x) = 
\begin{cases}
5, & x < -3 \\
x + 1, & -3 \leq x < 0 \\
-x^2 + 1, & x \geq 0
\end{cases}
\]

### Step-by-step breakdown:

1. **For \( x < -3 \):**
   \[
   f(x) = 5
   \]
   This is a constant function. For values of \( x \) less than \( -3 \), the graph will be a horizontal line at \( y = 5 \) up to \( x = -3 \), not including \( x = -3 \).

2. **For \( -3 \leq x < 0 \):**
   \[
   f(x) = x + 1
   \]
   This is a linear function with a slope of 1 and a y-intercept of 1. The graph will be a straight line starting at \( x = -3 \), where \( f(-3) = -3 + 1 = -2 \), and ending just before \( x = 0 \), where \( f(0) = 0 + 1 = 1 \). The endpoint at \( x = 0 \) is excluded because of the strict inequality.

3. **For \( x \geq 0 \):**
   \[
   f(x) = -x^2 + 1
   \]
   This is a downward-facing parabola with vertex at \( (0, 1) \). The graph will start at \( (0, 1) \) and follow the parabola downward as \( x \) increases.

### Graph Sketch:

1. **For \( x < -3 \):** A horizontal line at \( y = 5 \) from \( x = -\infty \) to just before \( x = -3 \).
2. **For \( -3 \leq x < 0 \):** A straight line from \( (-3, -2) \) to \( (0, 1) \) (but excluding \( x = 0 \)).
3. **For \( x \geq 0 \):** A downward-facing parabola starting at \( (0, 1) \) and curving downward.

Would you like me to generate an image of the graph for you?

Here are some related questions for deeper understanding:

1. What are the domain and range of the function \( f(x) \)?
2. How does the continuity of the function change at the boundaries \( x = -3 \) and \( x = 0 \)?
3. What are the intercepts of the function with the axes?
4. Can you describe the behavior of the function as \( x \to -\infty \) and \( x \to \infty \)?
5. What is the vertex of the parabola in the third case \( f(x) = -x^2 + 1 \)?

**Tip:** In piecewise functions, carefully evaluate the continuity at the boundary points by checking the left-hand and right-hand limits.

Sketch the graph of the piecewise function provided in the image.

Learn how to sketch the graph of a piecewise function with constant, linear, and quadratic components. This detailed step-by-step solution breaks down the function into its individual cases, ensuring a clear understanding of graph behavior. Suitable for high school students in Grades 10-12.

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