https://file.aiquickdraw.com/mathbot/file/1725793103053ssu857p1.jpg

The function given in the image is a piecewise function:

\[
f(x) = 
\begin{cases} 
1 + x & \text{for } x \leq 3 \\
4 & \text{for } x > 3
\end{cases}
\]

### Steps to sketch the graph:

1. **For \( x \leq 3 \), the function is \( f(x) = 1 + x \):**
   - This is a straight line with a slope of 1 and a y-intercept of 1.
   - The line extends up to \( x = 3 \). At \( x = 3 \), the value of \( f(x) = 1 + 3 = 4 \).

2. **For \( x > 3 \), the function is \( f(x) = 4 \):**
   - This is a horizontal line at \( y = 4 \) starting from \( x = 3 \) (not including 3) and going to infinity for increasing \( x \).

3. **Graph behavior at \( x = 3 \):**
   - At \( x = 3 \), both the line from the first part and the constant line from the second part meet at \( f(3) = 4 \). Therefore, there is no discontinuity or break at this point, making the graph continuous.

### Key points to consider:
- From \( x = -\infty \) to \( x = 3 \), the graph is a straight line increasing with slope 1.
- For \( x > 3 \), the graph becomes a horizontal line at \( y = 4 \).
- The graph is continuous at \( x = 3 \) because both parts of the piecewise function agree at that point.

Would you like me to generate a sketch of the graph based on this explanation?

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Here are 5 questions to further explore this problem:

1. How would the graph change if the second part of the piecewise function was \( f(x) = 5 \) for \( x > 3 \)?
2. What happens to the graph if we introduce a discontinuity at \( x = 3 \) (e.g., change \( f(3) \))?
3. How do we determine the slope of the linear part of the graph based on the equation \( f(x) = 1 + x \)?
4. What role does the boundary \( x = 3 \) play in defining the overall shape of the graph?
5. How would we describe the domain and range of this function?

### Tip:
When sketching piecewise functions, always check for continuity at the points where the function definition changes.

Sketch the graph of the following piecewise function: f(x) = {1 + x for x ≤ 3, 4 for x > 3}.

Learn how to graph the piecewise function f(x) = {1 + x for x ≤ 3, 4 for x > 3}. This problem demonstrates the behavior of a piecewise function with a linear segment and a constant segment, including the concept of continuity. Suitable for students in Grades 10-12.

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