https://file.aiquickdraw.com/mathbot/file/1726027927288g6ektkgp.jpg

Here is the graph of the function \( f(x) \). As we discussed earlier, the function is continuous at \( x = -2 \) because the left-hand limit, right-hand limit, and the function's value at \( x = -2 \) all agree at \( -4 \), which can be seen on the graph (marked by the red dot at \( (-2, -4) \)).

Would you like further clarification on this, or do you have any other questions?

Here are 5 related questions for further exploration:
1. How do you determine if a piecewise function is differentiable at the boundary point?
2. What is the derivative of \( f(x) \) for each piece of this piecewise function?
3. Can a function be continuous but not differentiable at a certain point? If so, provide examples.
4. How would you modify this function to make it discontinuous at \( x = -2 \)?
5. How do limits play a role in determining continuity for more complex piecewise functions?

**Tip**: When analyzing continuity, always check the limits from both the left and right at any potential points of discontinuity.

Suppose that the function f is defined, for all real numbers, as follows:

f(x) = { x - 2 if x < -2; 3x + 2 if x ≥ -2 }.
Graph the function f. Then determine whether or not the function is continuous.

Learn how to graph the piecewise function f(x) = { x - 2 if x < -2; 3x + 2 if x ≥ -2 } and determine its continuity at the point x = -2. This problem covers key concepts in limits and continuity, suitable for students in Grades 11-12.

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