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Magazine Chapter 1: Problem 109
Graph the function \(f(x)=[|3 x|]\) using a graphing utility. State the domain and range.
Short Answer
Expert verified
The domain is all real numbers \((-\infty, \infty)\), and the range is non-negative integers \(\{0, 1, 2, 3, \ldots \}\).
Step by step solution
01
Understand the Function
The function given is \( f(x) = [|3x|] \), where \([| ullet |]\) denotes the greatest integer function or floor function. This means for any real number input \( a \), \([| a |]\) is the greatest integer less than or equal to \( a \).
02
Analyze the Domain
The function \( f(x) = [|3x|] \) takes any real number \( x \) as its input. Thus, the domain of the function is all real numbers, expressed as \( (-\infty, \infty) \).
03
Evaluate Points for the Range
Evaluate \( f(x) = [|3x|] \) at specific points to understand the pattern. For example, at \( x = 0 \), \( f(0) = [|0|] = 0 \). At any non-zero \( x \), like \( x = 1 \), \( f(1) = [|3|] = 3 \). For \( x = -1 \), \( f(-1) = [|-3|] = 3 \). Continue similar evaluations.
04
Notice the Step Pattern
Observe that \( f(x) \) produces integer outputs. Every 1/3 interval change in x results in an increase in output by 1, confirming a staircase pattern.
05
Define the Range
Since \( f(x) \) outputs all non-negative integers as shown in the evaluations, the range of the function is non-negative integers \( \{0, 1, 2, 3, \ldots \} \).
06
Graph with a Utility
Use a graphing tool to plot the function. Enter the function \( f(x) = [|3x|] \). The graph will display a staircase-like curve moving upward from the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Domain
The domain of a function includes all the possible input values (or 'x' values) that the function can accept. For the greatest integer function,
denoted as \([|a|]\), it can take any real number 'a' as input. When we look at the function \(f(x) = [|3x|]\), we see that 'x' can
be any real number. That's because multiplying any real number by 3 still results in a real number. Therefore, the domain of this function is all real numbers,
which is represented as \((-\infty, \infty)\). This means there aren鈥檛 any restrictions on the input values. You can plug in negative numbers, zero, and positive numbers without
any issues.
denoted as \([|a|]\), it can take any real number 'a' as input. When we look at the function \(f(x) = [|3x|]\), we see that 'x' can
be any real number. That's because multiplying any real number by 3 still results in a real number. Therefore, the domain of this function is all real numbers,
which is represented as \((-\infty, \infty)\). This means there aren鈥檛 any restrictions on the input values. You can plug in negative numbers, zero, and positive numbers without
any issues.
Function Range
The range of a function refers to all the possible output values (or 'y' values) that you can get after evaluating the function with permissible inputs. In the case of
the function \(f(x) = [|3x|]\), let's explore its range by looking at what happens when you multiply a number by 3 and apply the greatest integer function.
the function \(f(x) = [|3x|]\), let's explore its range by looking at what happens when you multiply a number by 3 and apply the greatest integer function.
- When you multiply by 3, the result can be any real number.
- Applying the greatest integer function means you take the largest integer less than or equal to that real number.
- As a result, the range of this function includes all non-negative integers.
Graphing Functions
Graphing functions involves plotting the input values against their corresponding output values to visually represent the behavior of a function. Using a graphing utility
for \(f(x) = [|3x|]\) reveals a specific path. This is because every 1/3 interval of 'x' causes the output value to jump an entire integer step up or down, depending on the input.
which is a hallmark of the greatest integer functions.
for \(f(x) = [|3x|]\) reveals a specific path. This is because every 1/3 interval of 'x' causes the output value to jump an entire integer step up or down, depending on the input.
- When you draw the graph, instead of a smooth curve, you get step-like structures, since the function is a step function.
- Each 'step' corresponds to one unit of the output increasing with large constant intervals \([x, x+1)\) on the x-axis.
which is a hallmark of the greatest integer functions.
Step Function
A step function is a piecewise function that increases or decreases abruptly at certain intervals, representing constant output values within specific input intervals. The function \(f(x) = [|3x|]\) is such a function.
The key characteristics of a step function are:
of similar functions in different scenarios. This behavior is quite different from linear or quadratic functions, which change more continuously, providing a unique learning experience.
The key characteristics of a step function are:
- It has a constant value over each interval until the input crosses a new whole number defined by the function design (like every 1/3 interval here).
- There is a sudden jump to another constant value when moving from one interval to the next,
mirroring a staircase with steps of equal width.
of similar functions in different scenarios. This behavior is quite different from linear or quadratic functions, which change more continuously, providing a unique learning experience.
