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Magazine Chapter 13: Problem 42
Graph each step function. \(f(x)=[x]+1\)
Short Answer
Expert verified
Graph horizontal steps at every integer, shifting up by 1.
Step by step solution
01
- Understand the function
The given function is a step function written as \( f(x) = \lfloor x \rfloor + 1 \). Here, \( \lfloor x \rfloor \) denotes the floor function, which returns the greatest integer less than or equal to \( x \).
02
- Identify the behavior of the floor function
The floor function \( \lfloor x \rfloor \) steps down to the nearest integer. For example, \( \lfloor 2.3 \rfloor = 2 \) and \( \lfloor -1.7 \rfloor = -2 \). For every integer value \( n \), \( \lfloor n \rfloor = n \).
03
- Add 1 to the floor function
Add 1 to the result of the floor function. For any value of \( x \), we have \( f(x) = \lfloor x \rfloor + 1 \). For example, \( f(2.3) = \lfloor 2.3 \rfloor + 1 = 2 + 1 = 3 \) and \( f(-1.7) = \lfloor -1.7 \rfloor + 1 = -2 + 1 = -1 \).
04
- Plot the function piecewise
For each integer interval \( [n, n+1) \), where \( n \) is an integer, \( f(x) \) takes on a constant value of \( n+1 \). For instance, on the interval \( [0, 1) \), \( f(x) = \lfloor x \rfloor + 1 = 0 + 1 = 1 \). On the interval \( [1, 2) \), \( f(x) = \lfloor x \rfloor + 1 = 1 + 1 = 2 \). Continue this process for subsequent intervals.
05
- Draw the graph
Draw horizontal line segments for each interval. For \( x \) in the interval \( [n, n+1) \), draw a horizontal line segment at height \( n + 1 \). Do this for several values of \( n \) to illustrate the step function behavior. Make sure to use open dots at \( x = n+1 \) and closed dots at \( x = n \) to indicate the values that \( f(x) \) assumes within each interval.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Floor Function
The floor function, denoted as \( \lfloor x \rfloor \), is a mathematical function that returns the greatest integer less than or equal to \( x \). Think of it as rounding down to the nearest whole number. For instance:
- If \( x = 2.3 \), then \( \lfloor 2.3 \rfloor = 2 \).
- If \( x = -1.7 \), then \( \lfloor -1.7 \rfloor = -2 \).
- If \( x \) is already an integer, like \( 3 \), then \( \lfloor 3 \rfloor = 3 \).
Piecewise Function
A piecewise function is a function composed of multiple sub-functions, each applying to different intervals of the domain. For the given function \( f(x) = \lfloor x \rfloor + 1 \), it's piecewise because it takes different values depending on the interval of \( x \):
- On the interval \( [0, 1) \), \( f(x) = 1 \).
- On the interval \( [1, 2) \), \( f(x) = 2 \).
- This pattern continues for subsequent intervals.
Integer Intervals
Integer intervals are segments between two integers. For example, the interval \( [n, n+1) \) includes all real numbers from \( n \) up to, but not including, \( n+1 \). In context of our function:
- For the interval \( [0, 1) \), \( f(x) = 1 \).
- For the interval \( [1, 2) \), \( f(x) = 2 \).
- And so forth.
Graphing Techniques
When graphing a step function like \( f(x) = \lfloor x \rfloor + 1 \), follow these graphing tips and techniques to clearly illustrate the step behavior:
- Identify the integer intervals where the function takes on constant values.
- Plot horizontal line segments for each interval. For example, for \( x \) in \( [0, 1) \), draw a line at height 1.
- Use open dots to indicate where the interval ends and the point is not included (e.g., at \( x=1 \) between \( [0, 1) \)).
- Use closed dots to indicate where the interval starts and the point is included (e.g., at \( x=0 \)).
- Repeat this process for multiple intervals to create the full graph.
