The graph provided shows a piecewise function $y = g(x)$, and I'll analyze its key features:

1. Intervals:

   • The function starts from $x = 0$ and extends to $x = 6$.
   • The graph consists of different segments, including linear parts and a curve.

2. Domain and Range:

   • Domain: The function appears to be defined for $0 \leq x \leq 6$.
   • Range: The range seems to vary from $0 \leq y \leq 4$.

3. Key Points and Behavior:

   • At $x = 0$, $g(0) = 0$.
   • The graph moves upward linearly and reaches a value of $g(1) = 3$, then stays constant from $x = 1$ to $x = 3$ with a value of 3.
   • At $x = 3$, there is a hollow dot (indicating a discontinuity), followed by a solid dot at the same point, implying $g(3)$ might be redefined.
   • From $x = 3$ to $x = 5$, the function increases to a peak (maximum) at $g(4) = 4$, before decreasing back to $g(5) = 2$.
   • Finally, the graph approaches $y = 0$ as $x$ approaches 6.

4. Discontinuity:

   • There seems to be a jump discontinuity at $x = 3$, where the function transitions from the flat portion at $y = 3$ to a different curve.

Questions:

1. What is the significance of the hollow and solid dots at $x = 3$?
2. How would you describe the continuity of the function across its domain?
3. What are the local maximum and minimum points of the function?
4. How does the slope change from $x = 1$ to $x = 3$?
5. Can the function be expressed as a piecewise function with separate rules for different intervals?

Tip:

Always look for key points like maxima, minima, and discontinuities in a graph to understand its behavior better.