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To determine the values of \( a \) and \( b \) such that the given function is continuous at every point, we need to check the continuity at the points where the piecewise function changes, which are at \( x = -3 \) and \( x = 3 \).

The function is defined as:
\[
f(x) = 
\begin{cases} 
-8 & \text{for } x \leq -3 \\
ax - b & \text{for } -3 < x < 3 \\
13 & \text{for } x \geq 3
\end{cases}
\]

### Step 1: Continuity at \( x = -3 \)

For the function to be continuous at \( x = -3 \), the left-hand limit and the right-hand limit must both equal the function value at \( x = -3 \).

- The left-hand limit as \( x \to -3^- \) is the value of \( f(x) \) when \( x \leq -3 \), which is \( -8 \).
- The right-hand limit as \( x \to -3^+ \) is the value of the middle piece \( ax - b \) evaluated at \( x = -3 \).

Thus, for continuity at \( x = -3 \), we need:
\[
\lim_{x \to -3^-} f(x) = \lim_{x \to -3^+} f(x) = f(-3)
\]
\[
-8 = a(-3) - b
\]
\[
-8 = -3a - b
\]
This gives us the equation:
\[
3a + b = 8 \quad \text{(Equation 1)}
\]

### Step 2: Continuity at \( x = 3 \)

For the function to be continuous at \( x = 3 \), the left-hand limit and the right-hand limit must both equal the function value at \( x = 3 \).

- The left-hand limit as \( x \to 3^- \) is the value of \( ax - b \) evaluated at \( x = 3 \).
- The right-hand limit as \( x \to 3^+ \) is the value of \( f(x) \) when \( x \geq 3 \), which is 13.

Thus, for continuity at \( x = 3 \), we need:
\[
\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3)
\]
\[
a(3) - b = 13
\]
\[
3a - b = 13 \quad \text{(Equation 2)}
\]

### Step 3: Solving the system of equations

We now have the following system of equations:
1. \( 3a + b = 8 \)
2. \( 3a - b = 13 \)

We can solve this system by adding both equations:
\[
(3a + b) + (3a - b) = 8 + 13
\]
\[
6a = 21
\]
\[
a = \frac{21}{6} = 3.5
\]

Now substitute \( a = 3.5 \) into one of the original equations. Using Equation 1:
\[
3(3.5) + b = 8
\]
\[
10.5 + b = 8
\]
\[
b = 8 - 10.5 = -2.5
\]

### Conclusion:

The function is continuous at every point if \( a = 3.5 \) and \( b = -2.5 \).

Would you like more details on any specific step, or do you have further questions?

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Here are 5 relative questions:
1. How do we define continuity for piecewise functions?
2. What are the conditions for a function to be continuous at a point?
3. How would the solution change if the function had different break points?
4. What happens if the function is discontinuous at one of the points?
5. How can we visualize the continuity of this function graphically?

**Tip:** To check the continuity of piecewise functions, always analyze the limits from both sides at the points where the function changes its definition.

For what values of a and b is the following function continuous at every x?

Determine the values of a and b such that the piecewise function is continuous for all x, including at x = -3 and x = 3. Learn key concepts like limits and continuity in this advanced algebra problem.

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