We are tasked with determining the range of the given function:

$y = 3|x - 1| + 3$

on the restricted domain $0 \leq x \leq 3$.

Step-by-Step Solution:

1. Analyze the Absolute Value Expression: $|x - 1|$ depends on whether $x$ is less than or greater than $1$:

   • If $x \geq 1$, then $|x - 1| = x - 1$.
   • If $x < 1$, then $|x - 1| = 1 - x$.

2. Piecewise Function Definition: The function can be written as:

   • For $x \geq 1$: $y = 3(x - 1) + 3 = 3x - 3 + 3 = 3x$,
   • For $x < 1$: $y = 3(1 - x) + 3 = 3 - 3x + 3 = -3x + 6$.

3. Calculate Values at Critical Points:

   • Domain: $0 \leq x \leq 3$. Critical points are at the domain endpoints $x = 0, 3$ and $x = 1$ (where the absolute value changes).

   At $x = 0$: Using $-3x + 6$, $y = -3(0) + 6 = 6$.

   At $x = 1$: The value of $y$ transitions to $y = 3(1) = 3$.

   At $x = 3$: Using $3x$, $y = 3(3) = 9$.

4. Determine the Range: The minimum $y$-value occurs at $x = 1$, and the maximum $y$-value occurs at $x = 3$: $\text{Range: } 3 \leq y \leq 9.$

Thus, the range is $a = 3$ and $b = 9$.

Would you like a graph or further clarification? Here are some related questions to consider:

1. How does the graph of an absolute value function like this behave geometrically?
2. What is the significance of critical points in determining a function's range?
3. How would the range change if the domain were extended or restricted further?
4. Can you find a general formula for the range of $y = a|x - h| + k$?
5. How would adding or removing the constant $+3$ affect the range?

Tip: Always check for transitions in piecewise functions when dealing with absolute values and restricted domains!