Part (a): Graph Interpretation

The graph of the piecewise function $h(x)$ is shown. It consists of two parts:

1. The linear function $h(x) = 3x + 4$ for $x \leq 0$ (in blue).
2. The quadratic function $h(x) = (x - 2)^2$ for $x > 0$ (in red).

Notice the open circle at the point $(0, 4)$, indicating that the function is not defined at this point for the quadratic part.

Part (b): Discontinuity

The function might be discontinuous where the two pieces meet, which is at $x = 0$.

To check for discontinuity, let's evaluate the left-hand limit, right-hand limit, and the function value at $x = 0$.

1. Left-hand limit: This is the limit of the linear part as $x$ approaches 0 from the left. $\lim_{x \to 0^-} h(x) = \lim_{x \to 0^-} (3x + 4) = 3(0) + 4 = 4$

2. Right-hand limit: This is the limit of the quadratic part as $x$ approaches 0 from the right. $\lim_{x \to 0^+} h(x) = \lim_{x \to 0^+} (x^2 - 4x + 4) = (0)^2 - 4(0) + 4 = 4$

3. Function value at $x = 0$: Since $h(x) = 3x + 4$ for $x \leq 0$, the value at $x = 0$ is: $h(0) = 3(0) + 4 = 4$

Since the left-hand limit, right-hand limit, and the function value at $x = 0$ are all equal, the function is continuous at $x = 0$. Therefore, there are no discontinuities.

Part (c): Limits at Points of Discontinuity

As determined in part (b), the function is continuous everywhere, so there are no points where we need to compute left-hand or right-hand limits specifically for discontinuity.

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Do you need further details or have any additional questions?

Here are 5 related questions you could explore:

1. How would the graph change if the quadratic part included $x = 0$?
2. What would happen if the two parts of the function did not meet at the same point?
3. How do you find discontinuities in more complex piecewise functions?
4. What role do limits play in determining the continuity of a function?
5. How can we algebraically prove continuity for other types of piecewise functions?

Tip: Always check both the limits and the function's value when assessing continuity at a piecewise-defined boundary point.