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Chapter 12: Problem 88

Graph the following greatest integer functions. $$f(x)=[2 x]$$

Short Answer

Expert verified
To graph \(f(x) = [2x]\), first understand that the function returns the greatest integer less than or equal to \(2x\). Select points within the domain, such as -2, -1, 0, 1, and 2, and evaluate the function at each point: \((-2, -4), (-1, -2), (0, 0), (1, 2),\) and \((2, 4)\). Plot the points on the graph and connect them with step-like lines. Note that the function is discontinuous at integer values of x when the product \(2x\) is an integer.

Step by step solution

01

Understanding the Greatest Integer Function

The greatest integer function, denoted as \([x]\), returns the greatest integer less than or equal to \(x\). For example, \([3.2]=3\), \([-1.5]=-2\), and \([2]=2\). Now let's analyze the given function: \(f(x)=[2x]\).
02

Select Points for the Domain

To graph the function, we'll select a few points within the domain. The domain could be any real number. Let's choose a few representative points: -2, -1, 0, 1, and 2.
03

Evaluate the Function at Each Point

Now, we'll plug in each chosen point into the function and evaluate the resulting value: 1. \(f(-2)=[2(-2)]=[-4]=-4\) 2. \(f(-1)=[2(-1)]=[-2]=-2\) 3. \(f(0)=[2(0)]=[0]=0\) 4. \(f(1)=[2(1)]=[2]=2\) 5. \(f(2)=[2(2)]=[4]=4\)
04

Plot the Points on the Graph

Now that we have the corresponding points, let's plot them on the graph: (-2, -4), (-1, -2), (0, 0), (1, 2), and (2, 4).
05

Draw the Function

To draw the greatest integer function, we'll connect our plotted points with step-like lines. Remember that the floor function always "steps down" to the greatest integer less than or equal to the value of \(2x\). This results in a series of horizontal lines that take a "step down" whenever the function reaches an integer value.
06

Check for Continuity

As we can observe, the greatest integer function is discontinuous at integer values of x when the product \(2x\) is an integer. The function displays jumps at these points, but it is continuous between those points. Now that we've plotted the points, evaluated the function at those points, and drawn the function, you have successfully graphed the greatest integer function \(f(x)=[2x]\).

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Key Concepts

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

Graphing
When graphing functions, it鈥檚 important to accurately represent how they behave. The greatest integer function, such as \([2x]\) in our case, provides a unique visual due to its step-like nature. To begin graphing, you should first
  • Understand what the function is doing (it transforms each input, or x, into the greatest integer less than or equal to twice of it).
  • Select a range of x-values, like -2 to 2.
  • Calculate the corresponding y-values using the function.
For example, for the point \((-2, -4)\), you evaluate \([2*(-2)]\) to obtain -4.
After plotting these calculated points on a coordinate plane, connect the dots with horizontal lines to reflect how the function takes steps at each integer point.
The essential aspect of graphing the greatest integer function lies in showing the abrupt steps or jumps accurately at the integer values.
Step Functions
Step functions are a particular type of piecewise function defined by constant value intervals. A classical example is the greatest integer function.
For \([2x]\), the function stays constant between integer values but jumps to the next level at each integer point.
Some characteristics of step functions include:
  • Constant on intervals: Between integers, the output remains the same, forming a flat step.
  • Jumps at integer points: The function value "steps" up or down sharply at integer inputs.
  • Defined completely by its behavior on intervals: Here, \([2x]\) is defined completely by how 2x behaves on every real number interval.
To visualize a step function correctly, ensure your graph reflects these constant intervals and sudden jumps accurately. The steps should begin at the point calculated, continuing horizontally until the next integer point is reached where it then jumps to the new value.
Discontinuous Functions
Discontinuity is a core aspect of step functions like the greatest integer function. Despite being continuous between integers, they have discontinuities precisely at those integer points.
This occurs because:
  • As x approaches any integer, \([2x]\) suddenly "jumps" to a different value.
  • These jumps indicate points where the function lacks a smooth, unbroken path.
What makes \([2x]\) discontinuous?
  • The non-existence of a limit at integer points: Right before an integer, the limit from the left approaches one value, while from the right, it鈥檚 different.
  • The abrupt change in height of the graph at integer values.
Thus, understanding and identifying discontinuity is crucial for properly interpreting these functions. When plotting such a graph, make it evident that at every integer step, the value shifts instantly, reinforcing the concept of discontinuity inherent to the greatest integer function.

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Most popular questions from this chapter

\(R(x)=18 x\) is the revenue function for the sale of \(x\) toasters, in dollars. The cost to manufacture \(x\) toasters, in dollars, is \(C(x)=15 x+2400\) a) Find the profit function, \(P(x)\), that describes the profit from the sale of \(x\) toasters. b) What is the profit from the sale of 800 toasters?

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Determine the domain of each function. A freight train travels at a constant speed of \(32 \mathrm{mph}\). The distance \(D,\) in miles, that the train travels after \(t\) hr is given by the function \(D(t)=32 t\) A. How far will the train travel after 3 hr? B. How far will the train travel after 8 hr? C. How long does it take for the train to travel 208 mi? D. Graph the function.

Determine the domain of each function. $$r(t)=-\sqrt{t}$$
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