To determine the values of $a$ and $b$ that make the function continuous for all values of $x$, we need to ensure that the function is continuous at the points where the piecewise function changes its form. These points are $x = -1$ and $x = 3$.

Step 1: Continuity at $x = -1$

At $x = -1$, the function transitions from $f(x) = 2$ to $f(x) = ax + b$. For continuity at $x = -1$, the value of the function at $x = -1$ must be the same from both pieces. So we set:

$f(x) = 2 \quad \text{at} \quad x = -1$ and $f(x) = ax + b \quad \text{at} \quad x = -1$

Substituting $x = -1$ into the second piece: $2 = a(-1) + b \quad \Rightarrow \quad -a + b = 2 \quad \text{(Equation 1)}$

Step 2: Continuity at $x = 3$

At $x = 3$, the function transitions from $f(x) = ax + b$ to $f(x) = -2$. For continuity at $x = 3$, the value of the function at $x = 3$ must be the same from both pieces. So we set:

$f(x) = -2 \quad \text{at} \quad x = 3$ and $f(x) = ax + b \quad \text{at} \quad x = 3$

Substituting $x = 3$ into the second piece: $-2 = a(3) + b \quad \Rightarrow \quad 3a + b = -2 \quad \text{(Equation 2)}$

Step 3: Solving the System of Equations

Now, we solve the system of two equations:

1. $-a + b = 2$
2. $3a + b = -2$

By subtracting Equation 1 from Equation 2, we eliminate $b$: $(3a + b) - (-a + b) = -2 - 2$ $3a + b + a - b = -4$ $4a = -4 \quad \Rightarrow \quad a = -1$

Substitute $a = -1$ into Equation 1: $-a + b = 2 \quad \Rightarrow \quad -(-1) + b = 2 \quad \Rightarrow \quad 1 + b = 2 \quad \Rightarrow \quad b = 1$

Final Answer:

The values of $a$ and $b$ that make the function continuous for all values of $x$ are: $a = -1 \quad \text{and} \quad b = 1$

Would you like to explore the next problem or need further clarification on this one?

Here are some related questions:

1. How do we check if a piecewise function is differentiable?
2. What does it mean for a function to be continuous at a point?
3. How can we solve for constants in other piecewise function problems?
4. What is the concept of left-hand and right-hand limits in continuity?
5. How do we handle piecewise functions with more than two pieces?

Tip: Always check both the left-hand and right-hand limits when solving for continuity at a point!