Chapter 1: Problem 65
The ceiling function, or smallest integer function, \(f(x)=\lceil x\rceil,\) gives the smallest integer greater than or equal to \(x\). Graph the ceiling function, for \(-3 \leq x \leq 3\)
Short Answer
Expert verified
Answer: The graph of the ceiling function \(f(x) = \lceil x \rceil\) within the range \(-3 \leq x \leq 3\) looks like a series of horizontal lines that represent steps. There are discontinuous points at each integer value of x where the function jumps to the next integer value.
Step by step solution
01
Understand the Ceiling Function
The ceiling function \(\lceil x \rceil\) assigns to each number \(x\) the smallest integer greater than or equal to \(x\). This means that if \(x\) is already an integer, it maps to itself, and if it is a non-integer, it maps to the next integer in the positive direction.
For example, \(\lceil 3.1 \rceil = 4\) and \(\lceil -1.8 \rceil = -1\).
02
Create a Table of Values
Now we will create a table of values for the function \(f(x) = \lceil x \rceil\) for the given range \(-3 \leq x \leq 3\). Note that we will only consider the end-points (integers) and any necessary intermediate points.
- \(\lceil -3 \rceil = -3\)
- \(\lceil -2.999 \rceil = -2\)
- \(\lceil -2\rceil = -2\)
- \(\lceil -1.999 \rceil = -1\)
- \(\lceil -1 \rceil = -1\)
- \(\lceil -0.999 \rceil = 0\)
- \(\lceil 0 \rceil = 0\)
- \(\lceil 0.001 \rceil = 1\)
- \(\lceil 1 \rceil = 1\)
- \(\lceil 1.001 \rceil = 2\)
- \(\lceil 2 \rceil = 2\)
- \(\lceil 2.001 \rceil = 3\)
- \(\lceil 3 \rceil = 3\)
03
Plot the Function on the Graph
Based on the table of values, plot the points on a graph within the given range of \(-3 \leq x \leq 3\). Since the ceiling function is a step function, the graph will have discontinuous lines that resemble steps. The function is constant over an interval that includes all numbers between consecutive integers, but it jumps from one value to another at the integer points.
To draw the graph, plot the following segments:
1. \((−3,−3)\) to just before \((-2, -3)\). This segment must be a horizontal line.
2. \((−2,−2)\) to just before \((-1, -2)\). This segment must be a horizontal line.
3. \((−1,−1)\) to just before \((0, -1)\). This segment must be a horizontal line.
4. \((0,0)\) to just before \((1, 0)\). This segment must be a horizontal line.
5. \((1,1)\) to just before \((2, 1)\). This segment must be a horizontal line.
6. \((2,2)\) to just before \((3, 2)\). This segment must be a horizontal line.
7. At switching points, place an open circle on the left side of the switching point and a closed circle on the right side of the switching point. For example, at \(x=0\), there should be an open circle on the left and a closed circle on the right.
After completing these steps, you should have a step-like graph of the ceiling function \(f(x) = \lceil x \rceil\) for the range \(−3 \leq x \leq 3\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Step function
In mathematics, a step function is a type of function that displays changes in its outputs in a piecewise constant manner, similar to how actual stairs transition between steps. The ceiling function is a perfect example of a step function. The idea behind step functions is that for a range of input values, the function retains a constant output, and only changes or "steps" occur at certain points.
- Imagine walking up a staircase: you stay on one step for a while before moving up to the next step. Similarly, this function stays on the same output until the input hits an integer value.
- When graphing a step function like the ceiling function, you will notice that it has a series of flat, horizontal lines that abruptly "jump" to a higher level at specific points.
Graphing functions
Graphing functions is an essential skill in understanding how functions behave visually. When dealing with functions like the ceiling function, it's important to plot these functions correctly to observe their behavior. Graphing a ceiling function involves plotting each of its constant segments and ensuring that jumps are accurately represented.
- Start by creating a table of values which helps to establish specific points that the function will pass through.
- Next, transfer these consistent interval segments from the table onto a graph. From -3 to just before -2, for example, the output is -3, and so forth.
- Each segment is represented by a horizontal line. The function itself doesn’t include points at integer switches immediately until the next higher integer value. This is why we use open circles for the beginning of an interval and closed circles for the end.
Piecewise functions
A piecewise function is a function that is defined by multiple sub-functions, each applying to a certain interval of the main function's domain. The transition from one sub-function to another occurs at defined points. In the case of the ceiling function, each segment (piece) of the graph represents one piece of the piecewise function.
- The ceiling function applies a specific rule to each piece, i.e., giving the smallest integer greater than or equal to the value of the function.
- The formula for a piecewise function for a ceiling function could be represented focusing on when the function changes its rules at integer boundary points.
- The changes happening at integers make it clear where the function switches from one piece to another, perfectly reflected in the graph's step-like segments.
Discrete mathematics
Discrete mathematics involves the study of mathematical structures that are fundamentally discrete rather than continuous. In simpler terms, it deals with countable elements and often involves integers. The ceiling function is straightforwardly aligned with the principles of discrete mathematics.
- This function captures how mathematics handles discrete (non-continuous) data by jumping from one integer output to another.
- The ceiling function illustrates a scenario where the inputs can take any decimal form, but the outputs always snap to the nearest upper integer, reinforcing its role in discrete math.
- Understanding discrete functions like the ceiling is key in fields such as computer science, where discrete values are crucial for coding and algorithms.