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Here is the graph of the piecewise function \( h(x) \):

- For \( x < 3 \), the function is \( h(x) = \frac{2}{x-1} \), shown in blue.
- For \( x \geq 3 \), the function is \( h(x) = \frac{2}{3}x - 4 \), shown in orange.
- The vertical dashed green line indicates \( x = 3 \), where the function changes its behavior.

The graph shows a discontinuity at \( x = 3 \) because the values of the two pieces of the function don't meet at this point.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the limit of \( h(x) \) as \( x \) approaches 3 from the left and right?
2. How can you determine if the piecewise function is continuous at \( x = 3 \)?
3. What does the vertical asymptote at \( x = 1 \) for \( x < 3 \) mean in terms of behavior of the function?
4. How would you modify this graph to shift the discontinuity or the asymptote?
5. How can you solve for points of intersection between the two parts of the function?

**Tip:** Always check for discontinuities in piecewise functions by comparing the limits from both sides of the boundary point.

Learn how to graph a piecewise function involving a rational expression and a linear equation. This problem demonstrates key concepts like continuity, limits, and the presence of a discontinuity at the point x = 3. Suitable for students in Grades 10-12.

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