Graph showing a piecewise function with discontinuities at integer values of x, indicated by open and closed circles.

The image shows a graph of a piecewise function plotted on the coordinate plane. The function is represented by horizontal line segments, each with open and closed circles at the endpoints, indicating whether the endpoint is included or excluded in the domain.

Let’s analyze this step-by-step.

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Step 1: Understand the Graph

We observe:

- Each segment is horizontal, meaning the function has constant values over intervals.
- Open circles (○) indicate the point is not included in the interval.
- Closed circles (●) indicate the point is included.
- The function is defined for certain intervals of $ x $, and we can determine its value from the y-coordinate of the segment.

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Step 2: Identify Each Segment

We’ll go from left to right and describe each segment.

#### 1. Leftmost segment:
- From $ x = -4 $ to $ x = -3 $
- At $ x = -4 $: closed circle → included
- At $ x = -3 $: open circle → not included
- Value: $ y = -4 $
- So, $ f(x) = -4 $ for $ x \in [-4, -3) $

#### 2. Next segment:
- From $ x = -3 $ to $ x = -2 $
- At $ x = -3 $: closed circle → included
- At $ x = -2 $: open circle → not included
- Value: $ y = -3 $
- So, $ f(x) = -3 $ for $ x \in [-3, -2) $

#### 3. Next segment:
- From $ x = -2 $ to $ x = -1 $
- At $ x = -2 $: closed circle → included
- At $ x = -1 $: open circle → not included
- Value: $ y = -2 $
- So, $ f(x) = -2 $ for $ x \in [-2, -1) $

#### 4. Next segment:
- From $ x = -1 $ to $ x = 0 $
- At $ x = -1 $: closed circle → included
- At $ x = 0 $: open circle → not included
- Value: $ y = -1 $
- So, $ f(x) = -1 $ for $ x \in [-1, 0) $

#### 5. Next segment:
- From $ x = 0 $ to $ x = 1 $
- At $ x = 0 $: closed circle → included
- At $ x = 1 $: open circle → not included
- Value: $ y = 0 $
- So, $ f(x) = 0 $ for $ x \in [0, 1) $

#### 6. Next segment:
- From $ x = 1 $ to $ x = 2 $
- At $ x = 1 $: closed circle → included
- At $ x = 2 $: open circle → not included
- Value: $ y = 1 $
- So, $ f(x) = 1 $ for $ x \in [1, 2) $

#### 7. Next segment:
- From $ x = 2 $ to $ x = 3 $
- At $ x = 2 $: closed circle → included
- At $ x = 3 $: open circle → not included
- Value: $ y = 2 $
- So, $ f(x) = 2 $ for $ x \in [2, 3) $

#### 8. Rightmost segment:
- From $ x = 3 $ to $ x = 4 $
- At $ x = 3 $: closed circle → included
- At $ x = 4 $: open circle → not included
- Value: $ y = 3 $
- So, $ f(x) = 3 $ for $ x \in [3, 4) $

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Step 3: Write the Piecewise Function

Putting it all together:

$$
f(x) =
\begin{cases}
-4, & -4 \leq x < -3 \\
-3, & -3 \leq x < -2 \\
-2, & -2 \leq x < -1 \\
-1, & -1 \leq x < 0 \\
0, & 0 \leq x < 1 \\
1, & 1 \leq x < 2 \\
2, & 2 \leq x < 3 \\
3, & 3 \leq x < 4 \\
\end{cases}
$$

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Step 4: Observe the Pattern

Notice that:
- The function increases by 1 at each integer step.
- It's defined on intervals of length 1.
- The output $ f(x) $ is equal to $ x $ only if $ x $ is an integer and $ x \in [-4, 4) $, but wait — let’s check:

Actually, at $ x = -4 $: $ f(-4) = -4 $ ✔
At $ x = -3 $: $ f(-3) = -3 $ ✔
...
At $ x = 3 $: $ f(3) = 3 $ ✔
But at $ x = 4 $: not defined (open circle)

So for integers $ x \in \{-4, -3, -2, -1, 0, 1, 2, 3\} $, $ f(x) = x $

And the function is constant between these points.

This looks like the floor function shifted?

Wait — no. Actually, this is very similar to the greatest integer function, but not quite.

Wait — actually, notice:

For $ x \in [n, n+1) $, $ f(x) = n $, where $ n = -4, -3, ..., 3 $

So this is exactly the floor function!

Because $ \lfloor x \rfloor = n $ when $ x \in [n, n+1) $

So for $ x \in [-4, 4) $, $ f(x) = \lfloor x \rfloor $

✔ Yes! This function is the floor function restricted to $ [-4, 4) $

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Final Answer:

The graph represents the floor function over the interval $ [-4, 4) $. That is,

$$
f(x) = \lfloor x \rfloor \quad \text{for} \quad x \in [-4, 4)
$$

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Summary:

- The function is piecewise constant.
- Each horizontal segment corresponds to $ f(x) = n $ on $ [n, n+1) $
- The pattern matches $ f(x) = \lfloor x \rfloor $
- Domain: $ [-4, 4) $
- Range: $ \{-4, -3, -2, -1, 0, 1, 2, 3\} $

✔ Solution: The function shown is $ f(x) = \lfloor x \rfloor $ for $ x \in [-4, 4) $.