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Chapter 1: Problem 33

Sketch the graph of the function. $$g(x)=[[ x]] -1$$

Short Answer

Expert verified
The graph of the function \(g(x) = [[x]] -1\) should look like a staircase descending to the right, with steps at each integer value of \(x\), starting at \(y = -1\) for \(x = 0\) and descending by 1 unit at every subsequent integer \(x\). Jumps occur right at each integer \(x\), where the function moves from \(y = n - 1\) to \(y = n - 2\), \(n\) being the integer value of \(x\).

Step by step solution

01

Understand the Nature of the Function

The function \(g(x) = [[x]] - 1\) is a step function, because the greatest integer function, denoted [[x]], rounds \(x\) down to the nearest integer. Therefore, the function will have a constant value on each interval between integer values of \(x\), jumping to a new value at each integer.
02

Determine Function Values for Specific Intervals

Let's work out how the function behaves within specific intervals. For example, for \(x\) in \((0,1)\), \(g(x)=[[x]]-1=[[0]]-1=-1\). Following the same rule, for \(x\) in \((-1,0)\), \(g(x)=[[-1]]-1=-2\). Thus, we can determine the function values for each interval based off this logic.
03

Sketch the Graph

To sketch the function graph, use function values calculated in Step 2. The graph will be a sequence of horizontal line segments. For instance, between \(x = 0\) and \(x = 1\) (not inclusive), the function value will remain at -1, thus a horizontal line segment will be drawn at \(y = -1\). Likewise, between \(x = -1\) and \(x = 0\) (not inclusive), the function value will be -2, and a horizontal line segment will be plotted at \(y = -2\). The graph will exhibit jumps at each integer value of \(x\), where the function 'jumps' to its next value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Integer Function
The greatest integer function, often symbolized as \( [[x]] \), is a mathematical function that returns the largest integer less than or equal to a given real number \( x \). For example:
  • When \( x = 2.7 \), \( [[x]] = 2 \) because 2 is the largest integer less than or equal to 2.7.
  • Likewise, if \( x = -1.3 \), \( [[x]] = -2 \) because -2 is the integer part that is less than or equal to -1.3.
The function produces a step pattern due to its piecewise constant nature. The value stays constant over open intervals between consecutive integers and changes abruptly at integer points. This unique behavior makes graphs involving the greatest integer function appear as a series of flat steps.
Graph Sketching
Sketching a graph of a function like \( g(x) = [[x]] - 1 \) involves understanding how the function behaves over different intervals. To do this:
  • Identify the distinct intervals between integers where the function takes constant values.
  • Calculate the function value for each interval using the greatest integer function. For instance, in the interval \((0, 1)\), the expression reduces to \( g(x) = 0 - 1 = -1 \).
  • Draw horizontal line segments to represent each constant interval value.
The function will look like a series of horizontal steps on the graph, with each step beginning at a new integer. At these integers, the function 'jumps' to its next value. Make sure the jump is clearly depicted where the open circle denotes a value excluded from an interval.
Piecewise Functions
Piecewise functions are defined by different expressions depending on the input interval. The function \( g(x) = [[x]] - 1 \) is a piecewise constant function because:
  • It uses intervals to define different constant values across its domain. For example, from \(-1 \lt x \lt 0 \), \( g(x) = -2 \), and from \( 0 \lt x \lt 1 \), \( g(x) = -1 \).
  • It experiences abrupt transitions or 'jumps' at the boundaries, typical of piecewise definitions.
Learning about piecewise functions helps you understand how a single function might behave differently over its domain. This is especially important in applications like programming and physics, where systems often change state abruptly. Visualizing each piece of a piecewise function can also simplify complex problems by breaking them into manageable chunks.

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