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Let's tackle question 8 step by step.

We are given the piecewise function:

\[
p(x) =
\begin{cases} 
3 - 2g(x) & \text{if } x < 3, \\
5k + f(x) & \text{if } x \geq 3,
\end{cases}
\]
and we are asked to find the value of \( k \) that makes \( p(x) \) continuous at \( x = 3 \).

### Conditions for Continuity at \( x = 3 \):
To ensure that \( p(x) \) is continuous at \( x = 3 \), the left-hand limit (as \( x \to 3^- \)) must equal the right-hand limit (as \( x \to 3^+ \)), and they both must equal \( p(3) \).

1. **Left-hand limit \( \lim_{x \to 3^-} p(x) \):**
   For \( x < 3 \), \( p(x) = 3 - 2g(x) \). So, we need to evaluate this at \( x = 3^- \).

   From the graph of \( g(x) \), observe the value of \( g(3) \). At \( x = 3 \), the value of \( g(x) \) is \( g(3) = 2 \).

   Therefore, the left-hand limit is:
   \[
   \lim_{x \to 3^-} p(x) = 3 - 2g(3) = 3 - 2(2) = 3 - 4 = -1.
   \]

2. **Right-hand limit \( \lim_{x \to 3^+} p(x) \):**
   For \( x \geq 3 \), \( p(x) = 5k + f(x) \). So, we need to evaluate this at \( x = 3^+ \).

   From the graph of \( f(x) \), observe the value of \( f(3) \). At \( x = 3 \), the value of \( f(x) \) is \( f(3) = -2 \).

   Therefore, the right-hand limit is:
   \[
   \lim_{x \to 3^+} p(x) = 5k + f(3) = 5k + (-2) = 5k - 2.
   \]

3. **Set the left-hand limit equal to the right-hand limit:**
   For continuity, we need:
   \[
   -1 = 5k - 2.
   \]
   Solving for \( k \):
   \[
   5k = -1 + 2 = 1,
   \]
   \[
   k = \frac{1}{5}.
   \]

### Conclusion:
The value of \( k \) that makes \( p(x) \) continuous at \( x = 3 \) is \( k = \frac{1}{5} \).

Would you like further details or explanations? 

Here are 5 related questions for deeper understanding:
1. What does it mean for a function to be continuous at a point?
2. How would the solution change if the piecewise function had a different form for \( x < 3 \)?
3. What is the importance of evaluating both the left-hand and right-hand limits for continuity?
4. Can you explain how the graphs of \( f(x) \) and \( g(x) \) influence the limits in this problem?
5. How would we solve a similar problem if the graphs of \( f(x) \) and \( g(x) \) had discontinuities at \( x = 3 \)?

**Tip:** When solving continuity problems, always check the values of the function and its limits on both sides of the point of interest.

Solve question 8: Let p(x) be the piecewise defined function: p(x) = { 3 - 2g(x), x < 3; 5k + f(x), x ≥ 3. Find the value of k that makes p continuous at x = 3.

Find the value of k that makes the piecewise function p(x) continuous at x = 3. Learn to apply the Continuity Theorem by evaluating left-hand and right-hand limits. Suitable for Grades 10-12.

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