Problem 19 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. $$ f(x)=-[x] $$

Step-by-Step Solution

Answer

The graph consists of descending horizontal steps, each at integer values lower by one unit.

Step 1: Understand the Function

The function given is \( f(x) = -[x] \), where \([x]\) is the greatest integer function (denoted as floor function). \([x]\) gives the greatest integer less than or equal to \( x \). Applying the negative sign outside will invert the steps vertically. We need to focus on how the behavior of the floor function changes with respect to \( x \).

Step 2: Determine Key Points

Evaluate the function at various points around key integer values. For values between two integers, the floor function takes the lower integer value. For example, for \( x = 0 \), \( f(0) = -0 = 0 \); for \( x = 0.5 \), \( f(0.5) = -0 = 0 \); and right before \( x = 1 \), such as \( x = 0.99 \), \( f(0.99) = -0 = 0 \). At \( x = 1 \), \( f(1) = -1 \).

Step 3: Plot Points and Analyze Behavior

Plot the calculated points on the coordinate system:- At \( x = 0 \), \( f(x) = 0 \).- For \( 0 < x < 1 \), \( f(x) = 0 \).- At \( x = 1 \), \( f(x) = -1 \).- For \( 1 < x < 2 \), \( f(x) = -1 \).- At \( x = 2 \), \( f(x) = -2 \).This pattern continues.

Step 4: Sketch the Graph

The graph is composed of horizontal segments or "steps," each starting at an integer point \( (x, -x) \) and extending leftwards. Each segment between \( x = n \) and \( x = n+1 \) has the function value \( -n \), creating a step of length 1 vertically moving down by whole integers as \( x \) increases.

Key Concepts

Floor Function

The floor function, denoted as \([x]\), is a mathematical function that returns the greatest integer less than or equal to a given number \(x\). This means it rounds down \(x\) to the nearest whole number. For instance:

• \([3.7] = 3\) because 3 is the greatest integer less than 3.7.
• \([-1.5] = -2\) because -2 is the greatest integer less than or equal to -1.5.

The floor function steps abruptly from one integer value to the next as \(x\) crosses an integer boundary.
This creates a series of "steps," which is crucial for understanding how it interacts with other functions, especially in graphing scenarios.

Step Function

Step functions are discontinuous functions that change at intervals, forming "steps" on a graph. The floor function, \([x]\), is a classic example. Its graph consists of flat horizontal lines that jump at each integer. Each line corresponds to an interval between integers.
When a step function like the floor function is negated, as in this exercise with \(-[x]\), the steps are inverted vertically. Instead of stair-stepping upward, the graph stair-steps downward. This is key when graphing the function \(f(x) = -[x]\). Every segment between integers \(n\) and \(n+1\) is at a constant height of \(-n\).

Graph Sketching

To sketch the graph of \(f(x) = -[x]\), start by plotting key points based on integer values of \(x\). First, consider the value at an integer \(x\) like 0, where \(f(0) = 0\). As \(x\) varies from 0 to just under 1, \(f(x)\) stays at 0. Once \(x\) reaches 1, the output jumps to -1.

• \(x = 0\), \(f(x) = 0\)
• \(0 < x < 1\), \(f(x) = 0\)
• \(x = 1\), \(f(x) = -1\)
• \(1 < x < 2\), \(f(x) = -1\)
• \(x = 2\), \(f(x) = -2\)

This sequence continues with each subsequent integer \(x\). The graph appears as steps moving downwards, with each segment's length exactly 1.

Integer Values Analysis

Analyzing integer values helps understand the behavior of \(f(x) = -[x]\). At each integer point, \(x = n\), the value of the function becomes \(-n\). For any \(x\) between two integers \(n\) and \(n+1\), the function value remains fixed at \(-n\), forming a horizontal line segment.
Consider the pattern: when \(x\) is exactly an integer, \(f(x)\) jumps to the next lower integer's negative. For instance:

• At \(x = 2\), \(f(2) = -2\).
• For \(2 < x < 3\), \(f(x) = -2\).

This jump repeats every integer step, forming a consistent, predictable pattern as \(x\) increases. Recognizing this pattern is central to accurately predicting and sketching the function's graph.