Question

The floor function, or greatest integer function, \(f(x)=\lfloor x\rfloor,\) gives the greatest integer less than or equal to \(x\) Graph the floor function, for \(-3 \leq x \leq 3\)

Short answer

Answer: The graph of the floor function within the interval \(-3 \leq x \leq 3\) appears as a step-like curve with vertical jumps at each integer value: ``` y-axis 3| +--o 2| +--o 1| +--o 0| +--o -1| +--o -2| +--o -3| +--o-------------------> x-axis ```

Step-by-step solution

Explain the floor function

The floor function, written as \(f(x)=\lfloor x \rfloor\), gives us the largest integer that is less than or equal to \(x\). For example, \(\lfloor 2.3 \rfloor = 2\), \(\lfloor -0.5 \rfloor = -1\), and \(\lfloor -3 \rfloor = -3\).

Determine the integer values within the given interval

The given interval is \(-3 \leq x \leq 3\). The integer values within this interval are \(-3, -2, -1, 0, 1, 2,\) and \(3\).

Find the floor function values for each integer

Calculate the floor function values for the integers in the given interval: - \(\lfloor -3 \rfloor = -3\) - \(\lfloor -2 \rfloor = -2\) - \(\lfloor -1 \rfloor = -1\) - \(\lfloor 0 \rfloor = 0\) - \(\lfloor 1 \rfloor = 1\) - \(\lfloor 2 \rfloor = 2\) - \(\lfloor 3 \rfloor = 3\)

Graph the floor function within the interval

To graph the floor function over the given interval, plot the integer values on the x-axis and their corresponding floor function values on the y-axis. Then, draw the floor function as a step-like curve with vertical jumps at each integer value. The curve should be continuous from the left at each vertical jump, and have an open circle at the right end of each horizontal line segment. The graph should look like this: ``` y-axis 3| +--o 2| +--o 1| +--o 0| +--o -1| +--o -2| +--o -3| +--o-------------------> x-axis ```

Key concepts

Greatest Integer Function

The greatest integer function, often referred to as the floor function, is defined as \( f(x) = \lfloor x \rfloor \). This function returns the largest integer that is less than or equal to a given real number \( x \). For instance, if you consider the number \( 2.9 \), the greatest integer less than or equal to \( 2.9 \) is \( 2 \). Hence, \( \lfloor 2.9 \rfloor = 2 \). Similarly, for negative numbers like \( -0.2 \), the largest integer less than or equal to it is \( -1 \), so \( \lfloor -0.2 \rfloor = -1 \). The floor function steps down to the nearest lower integer unlike the ceiling function, which steps up to the nearest higher integer. This integral part of the floor function makes it incredibly helpful when dealing with discrete data or when rounding down is required.

Graphing Functions

Graphing functions helps us visualize their behavior on a coordinate plane. For the floor function, the task involves plotting integer values of \( x \) against their corresponding floor function values on the \( y \)-axis. When graphing the floor function between the interval \( -3 \leq x \leq 3 \), the graph exhibits a series of horizontal line segments or "steps."

• At each integer \( x \), the function value corresponds to \( x \) itself
• The function appears constant between two consecutive integers. For example, between \( x = 1 \) and \( x = 2 \),\( f(x) = 1 \)
• Each step has a closed circle at the integer value and an open circle just before the next integer, emphasizing the point where the function value steps down

The resulting graph of the floor function is a great visual demonstration of how the function operates step by step, illustrating its discontinuous nature. This step-like appearance helps learners better understand how the floor function works, especially in terms of continuity and jumps.

Step Functions

Step functions are types of functions characterized by segments of constant value. The floor function is a perfect example of a step function. Each "step" corresponds to a range of \( x \)-values over which the function remains constant, ending with a jump to a new value. The discontinuity is clearly represented on the graph by an open circle at each step's endpoint, highlighting where the function "steps down" to the next integer.Understanding step functions like the floor function involves recognizing that:

• They are not continuous. They jump at each integer value.
• They are simple to evaluate because the function’s value doesn’t change within a specific interval.
• Real-world applications can include scenarios such as ticket pricing, where the cost remains fixed until a certain threshold is crossed.

The step function’s discontinuity is what makes it stand out and is also what makes it quite useful in situations where gradual changes aren't necessary, or where rounding down to an integer value is essential. By studying such functions, students gain insight into how mathematical modeling of periodic data works.