If [] and {} represents the greatest integer function and fractional function then the graph of y = [x] and y = {x} respectively parallel to

Options

A. x-axis, y-axis
B. y-axis, x-axis
C. a-axis, y = x
D. y = 2x, y = x + 1

Answer

To solve the problem, we need to analyze the graphs of the greatest integer function \( y = [x] \) and the fractional part function \( y = \{x\} \).

### Step 1: Understanding the Greatest Integer Function \( y = [x] \)

The greatest integer function, denoted as \( [x] \), gives the largest integer less than or equal to \( x \).

- For example:
- \( [0.5] = 0 \)
- \( [1.2] = 1 \)
- \( [2.9] = 2 \)

#### Graph of \( y = [x] \):
- The graph consists of horizontal line segments at each integer value, starting from the integer and extending to the next integer, but not including it.
- For \( x \) in the interval \( [n, n+1) \), \( y = [x] = n \).
- The graph has open circles at \( n+1 \) and closed circles at \( n \).

### Step 2: Analyzing the Graph of \( y = [x] \)

- The graph of \( y = [x] \) is a series of horizontal line segments, which means it is parallel to the x-axis.
- The lines are horizontal because for any \( x \) in the interval \( [n, n+1) \), the output \( y \) remains constant at \( n \).

### Step 3: Understanding the Fractional Part Function \( y = \{x\} \)

The fractional part function, denoted as \( \{x\} \), gives the non-integer part of \( x \).

- For example:
- \( \{0.5\} = 0.5 \)
- \( \{1.2\} = 0.2 \)
- \( \{2.9\} = 0.9 \)

#### Graph of \( y = \{x\} \):
- The graph of \( y = \{x\} \) is a series of line segments that rise from \( 0 \) to \( 1 \) and then drop back to \( 0 \).
- For \( x \) in the interval \( [n, n+1) \), \( y = \{x\} = x - n \).
- The graph has open circles at integer values and starts from \( (n, 0) \) to \( (n+1, 1) \).

### Step 4: Analyzing the Graph of \( y = \{x\} \)

- The graph of \( y = \{x\} \) is a series of line segments that are slanted upwards, and each segment is parallel to the line \( y = x \).
- This is because the slope of the line segment is \( 1 \) in each interval \( [n, n+1) \).

### Conclusion

1. The graph of \( y = [x] \) is parallel to the x-axis.
2. The graph of \( y = \{x\} \) is parallel to the line \( y = x \).

### Final Answer
- The graph of \( y = [x] \) is parallel to the x-axis.
- The graph of \( y = \{x\} \) is parallel to the line \( y = x \).