https://file.aiquickdraw.com/mathbot/file/1727272073486z4hbhefu.jpg

The image shows a piecewise function and a condition for continuity, likely asking to find the values of parameters \( a \) and \( b \) such that the function is continuous. Here's the function:

\[
f(x) = \begin{cases} 
a \cdot x + 2 & \text{if } x \leq -3 \\
2 \cdot x + b & \text{if } -3 < x \leq 4 \\
25 & \text{if } x \geq 4
\end{cases}
\]

The problem appears to ask that the function be continuous at \( x = -3 \) and \( x = 4 \).

### Conditions for Continuity:
1. **At \( x = -3 \):**
   \[
   \lim_{x \to -3^-} f(x) = \lim_{x \to -3^+} f(x) = f(-3)
   \]
   From the first and second pieces of the function, we can equate these two:
   \[
   a \cdot (-3) + 2 = 2 \cdot (-3) + b
   \]
   This simplifies to:
   \[
   -3a + 2 = -6 + b
   \]
   \[
   b = -3a + 8 \quad \text{(Equation 1)}
   \]

2. **At \( x = 4 \):**
   \[
   \lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x) = f(4)
   \]
   From the second and third pieces of the function, we can equate these two:
   \[
   2 \cdot 4 + b = 25
   \]
   This simplifies to:
   \[
   8 + b = 25
   \]
   \[
   b = 17 \quad \text{(Equation 2)}
   \]

### Solving for \( a \) and \( b \):
- From **Equation 2**, we already know \( b = 17 \).
- Substituting \( b = 17 \) into **Equation 1**:
  \[
  17 = -3a + 8
  \]
  \[
  9 = -3a
  \]
  \[
  a = -3
  \]

### Final Answer:
- \( a = -3 \)
- \( b = 17 \)

Let me know if you want more details or have any questions!

Here are five related questions:
1. What is the general condition for continuity at a point for piecewise functions?
2. How do we approach finding limits in piecewise functions?
3. How would the solution change if the point of continuity was at a different value, say \( x = 2 \)?
4. What is the difference between one-sided limits and two-sided limits in continuity?
5. How would discontinuities appear graphically in such piecewise functions?

**Tip:** When dealing with continuity of piecewise functions, always focus on the boundary points where the different expressions of the function meet.

Find the values of a and b that make the function continuous.

Discover how to determine the coefficients a and b to ensure continuity in a piecewise function defined in intervals. This guide covers the steps of limit calculations and setting conditions for continuity at boundary points, ideal for university or advanced high school students studying calculus.

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