To find the range of the function $y = |x - 2| - 3$, we first need to understand how the absolute value function $|x - 2|$ behaves.

Step 1: Understand the absolute value function

The function $|x - 2|$ represents the distance of $x$ from 2 on the number line. This function always gives a non-negative value, so:

$|x - 2| \geq 0 \text{ for all } x.$

Step 2: Analyze the function

The given function is:

$y = |x - 2| - 3.$

Since $|x - 2| \geq 0$, we have:

$y = |x - 2| - 3 \geq 0 - 3 = -3.$

Thus, $y \geq -3$.

Step 3: Determine the range

The minimum value of $|x - 2|$ is 0 (when $x = 2$), so:

$y = |x - 2| - 3 = 0 - 3 = -3.$

As $|x - 2|$ can increase without bound (as $x$ moves away from 2 in either direction), $y$ can also increase without bound. Therefore, the possible values for $y$ are all real numbers greater than or equal to $-3$.

Conclusion

The range of the function $y = |x - 2| - 3$ is:

$\boxed{[-3, \infty)}.$

Would you like more details or have any questions?

Here are 5 related questions that might interest you:

1. How do you find the domain of an absolute value function?
2. What is the vertex of the function $y = |x - 2| - 3$?
3. How do transformations affect the graph of an absolute value function?
4. What are the intercepts of the function $y = |x - 2| - 3$?
5. How do you find the range of more complex functions involving absolute values?

Tip: Remember that the absolute value function creates a "V" shape graph, and understanding its transformations can help find ranges and other properties easily.