What is the range of the function y = I x - 2 I - 3?

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.

What is the range of the function y = |x - 2| - 3?

Discover how to find the range of the function y = |x - 2| - 3 with a step-by-step explanation. Learn key concepts like absolute value functions and how transformations affect the graph. Suitable for Grades 8-10.

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