Plot the following, where [.] denotes integer function.
`f(x)=[|x|-2]`

Answer

To plot the function \( f(x) = [|x| - 2] \), where \([.]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Understand the function
The function \( f(x) = [|x| - 2] \) involves the absolute value of \( x \) and then subtracts 2. The greatest integer function (or floor function) will take the result of \( |x| - 2 \) and round it down to the nearest integer.

### Step 2: Determine the intervals for \( f(x) \)
1. **Find when \( |x| - 2 \) is in specific ranges**:
- For \( f(x) = 0 \):
\[ 0 \leq |x| - 2 < 1 \implies 2 \leq |x| < 3 \]
This gives us two cases:
- \( 2 \leq x < 3 \) (for \( x \geq 0 \))
- \( -3 < x \leq -2 \) (for \( x < 0 \))

- For \( f(x) = 1 \):
\[ 1 \leq |x| - 2 < 2 \implies 3 \leq |x| < 4 \]
This gives us two cases:
- \( 3 \leq x < 4 \) (for \( x \geq 0 \))
- \( -4 < x \leq -3 \) (for \( x < 0 \))

- For \( f(x) = 2 \):
\[ 2 \leq |x| - 2 < 3 \implies 4 \leq |x| < 5 \]
This gives us two cases:
- \( 4 \leq x < 5 \) (for \( x \geq 0 \))
- \( -5 < x \leq -4 \) (for \( x < 0 \))

- For \( f(x) = 3 \):
\[ 3 \leq |x| - 2 < 4 \implies 5 \leq |x| < 6 \]
This gives us two cases:
- \( 5 \leq x < 6 \) (for \( x \geq 0 \))
- \( -6 < x \leq -5 \) (for \( x < 0 \))

- For \( f(x) = -1 \):
\[ -1 \leq |x| - 2 < 0 \implies 1 \leq |x| < 2 \]
This gives us two cases:
- \( 1 \leq x < 2 \) (for \( x \geq 0 \))
- \( -2 < x \leq -1 \) (for \( x < 0 \))

- For \( f(x) = -2 \):
\[ -2 \leq |x| - 2 < -1 \implies 0 \leq |x| < 1 \]
This gives us two cases:
- \( 0 \leq x < 1 \) (for \( x \geq 0 \))
- \( -1 < x \leq 0 \) (for \( x < 0 \))

### Step 3: Compile the intervals
Now we can summarize the intervals for \( f(x) \):
- \( f(x) = 0 \) for \( 2 \leq x < 3 \) and \( -3 < x \leq -2 \)
- \( f(x) = 1 \) for \( 3 \leq x < 4 \) and \( -4 < x \leq -3 \)
- \( f(x) = 2 \) for \( 4 \leq x < 5 \) and \( -5 < x \leq -4 \)
- \( f(x) = 3 \) for \( 5 \leq x < 6 \) and \( -6 < x \leq -5 \)
- \( f(x) = -1 \) for \( 1 \leq x < 2 \) and \( -2 < x \leq -1 \)
- \( f(x) = -2 \) for \( 0 \leq x < 1 \) and \( -1 < x \leq 0 \)

### Step 4: Plot the graph
1. **Draw the axes**: Create a Cartesian plane with x-axis and y-axis.
2. **Plot the intervals**:
- For each interval, mark the corresponding y-value on the graph.
- Use open circles for endpoints where the value is not included (e.g., \( 3 \) is not included in \( f(x) = 0 \)).
- Use closed circles for endpoints where the value is included (e.g., \( -2 \) is included in \( f(x) = -2 \)).

### Final Graph
The graph will consist of horizontal line segments at the respective y-values for the intervals found above.