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To solve Question 4, we need to determine whether the function \( p(x) = g(x) - f(x) \) is continuous at \( x = 4 \).

### Steps to solve the problem:

1. **Check continuity definition**: A function is continuous at \( x = a \) if:
   - \( \lim_{x \to a} p(x) \) exists,
   - \( p(a) \) exists,
   - \( \lim_{x \to a} p(x) = p(a) \).

2. **Calculate \( p(4) \)**: Using the graphs of \( f(x) \) and \( g(x) \), find the values of \( f(4) \) and \( g(4) \).
   - From the graph of \( f(x) \), \( f(4) = 0 \).
   - From the graph of \( g(x) \), \( g(4) = 4 \).
   Therefore, \( p(4) = g(4) - f(4) = 4 - 0 = 4 \).

3. **Find \( \lim_{x \to 4} p(x) \)**: Now, calculate the left-hand and right-hand limits of \( p(x) \) as \( x \) approaches 4.
   - **Left-hand limit** \( \lim_{x \to 4^-} p(x) \):
     - As \( x \) approaches 4 from the left, \( f(x) \to 0 \) and \( g(x) \to 4 \).
     - Thus, \( \lim_{x \to 4^-} p(x) = 4 - 0 = 4 \).
   
   - **Right-hand limit** \( \lim_{x \to 4^+} p(x) \):
     - As \( x \) approaches 4 from the right, \( f(x) \to 0 \) and \( g(x) \to 4 \).
     - Thus, \( \lim_{x \to 4^+} p(x) = 4 - 0 = 4 \).

4. **Conclusion**:
   - Both the left-hand and right-hand limits exist and are equal to 4, so \( \lim_{x \to 4} p(x) = 4 \).
   - Since \( p(4) = 4 \) and \( \lim_{x \to 4} p(x) = 4 \), the function \( p(x) \) is continuous at \( x = 4 \).

Thus, \( p(x) \) is continuous at \( x = 4 \).

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Do you want further details or explanations on any of the steps? 

Here are five related questions you might be interested in:

1. How do we define continuity for a piecewise function?
2. What if the left-hand and right-hand limits at \( x = 4 \) were not equal?
3. How would we find limits graphically if the function was not well-defined at some points?
4. What would it mean if \( p(x) \) were discontinuous at \( x = 4 \)?
5. Can a function be continuous but not differentiable at a point?

**Tip:** When checking for continuity at a point, always verify the limit

Let p(x) = g(x) - f(x). Is p(x) continuous at x = 4? Justify your answer.

Learn how to determine if the function p(x) = g(x) - f(x) is continuous at x = 4 using limits. Understand key concepts like left-hand and right-hand limits, and apply the definition of continuity in this step-by-step solution. Suitable for high school students in Grades 10-12.

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