Answer :
To determine which statement correctly describes the graph of the function \( f(x) = \lfloor x \rfloor - 2 \) on the interval \([0, 3)\), we need to analyze the behavior of \( f(x) \) within the sub-intervals \( [0, 1) \), \( [1, 2) \), and \( [2, 3) \).
### Step-by-Step Analysis
1. For the interval \( [0, 1) \):
- Within this range, the floor function \( \lfloor x \rfloor \) always returns 0 because \( x \) is non-negative and less than 1.
- So \( f(x) = \lfloor x \rfloor - 2 = 0 - 2 = -2 \).
2. For the interval \( [1, 2) \):
- Within this range, the floor function \( \lfloor x \rfloor \) always returns 1 because \( x \) is at least 1 but less than 2.
- So \( f(x) = \lfloor x \rfloor - 2 = 1 - 2 = -1 \).
3. For the interval \( [2, 3) \):
- Within this range, the floor function \( \lfloor x \rfloor \) always returns 2 because \( x \) is at least 2 but less than 3.
- So \( f(x) = \lfloor x \rfloor - 2 = 2 - 2 = 0 \).
### Conclusion
Given the descriptions of the steps, the correct statement can be identified:
- The steps are at \( y=-2 \) for \( 0 \leq x < 1 \),
- At \( y=-1 \) for \( 1 \leq x < 2 \),
- And at \( y=0 \) for \( 2 \leq x < 3 \).
Thus, the correct statement is:
```
The steps are at \( y=-2 \) for \( 0 \leq x < 1 \), at \( y=-1 \) for \( 1 \leq x < 2 \), and at \( y=0 \) for \( 2 \leq x < 3 \).
```
### Step-by-Step Analysis
1. For the interval \( [0, 1) \):
- Within this range, the floor function \( \lfloor x \rfloor \) always returns 0 because \( x \) is non-negative and less than 1.
- So \( f(x) = \lfloor x \rfloor - 2 = 0 - 2 = -2 \).
2. For the interval \( [1, 2) \):
- Within this range, the floor function \( \lfloor x \rfloor \) always returns 1 because \( x \) is at least 1 but less than 2.
- So \( f(x) = \lfloor x \rfloor - 2 = 1 - 2 = -1 \).
3. For the interval \( [2, 3) \):
- Within this range, the floor function \( \lfloor x \rfloor \) always returns 2 because \( x \) is at least 2 but less than 3.
- So \( f(x) = \lfloor x \rfloor - 2 = 2 - 2 = 0 \).
### Conclusion
Given the descriptions of the steps, the correct statement can be identified:
- The steps are at \( y=-2 \) for \( 0 \leq x < 1 \),
- At \( y=-1 \) for \( 1 \leq x < 2 \),
- And at \( y=0 \) for \( 2 \leq x < 3 \).
Thus, the correct statement is:
```
The steps are at \( y=-2 \) for \( 0 \leq x < 1 \), at \( y=-1 \) for \( 1 \leq x < 2 \), and at \( y=0 \) for \( 2 \leq x < 3 \).
```
