Chapter 7: Problem 23
\(f(x)=\llbracket 2 x \rrbracket+1\)
Short Answer
Expert verified
The function \( f(x) \) rounds down \( 2x \) to the nearest integer, then adds 1.
Step by step solution
01
Understand the Function
First, one needs to understand the function \( f(x)=\llbracket 2 x \rrbracket+1 \). The \( \llbracket 2 x \rrbracket \) part denotes the floor function so it will round \( 2x \) downwards to the nearest whole number and then we need to add 1 to this value.
02
Apply the function to a value of \( x \)
Choose a value of \( x \), substitute it into the \( 2x \) term inside the floor function, perform the floor operation (round down to the nearest whole number), and then add 1.
03
Generalize the process
This process of applying the function can be repeated for any real number for \( x \). It is noted that for half-integer values of \( x \), the rounding will still be downward, as this is the definition of the floor function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rounding
Rounding is an essential mathematical technique that adjusts a number to make it simpler and more manageable. It involves modifying a number to either its nearest higher or lower value based on certain rules. This is crucial when dealing with numbers that have many decimal places, as it allows us to simplify without losing meaningful accuracy.
There are different types of rounding strategies, including:
There are different types of rounding strategies, including:
- **Rounding up:** Adjusting a number to the next whole number.
- **Rounding down:** Adjusting a number to the previous whole number. This is also known as the **floor function**.
- **Rounding to nearest:** Rounding a number to the closest whole number based on specific rules.
Whole Number
A whole number is one of the simplest concepts in mathematics. Whole numbers include all non-negative numbers without any fractions or decimal parts. They consist of zero and all positive integers like 1, 2, 3, and so on.
Whole numbers are particularly significant when dealing with rounding because they are often the target of these operations. For example, in the floor function within our exercise, we aim to round a real number down to the nearest whole number.
Whole numbers are vital in both theoretical and practical mathematics as they form the basis for counting and simple arithmetic. They make it possible to perform operations like addition, subtraction, multiplication, and division without the complications that fractions or decimals bring.
Whole numbers are particularly significant when dealing with rounding because they are often the target of these operations. For example, in the floor function within our exercise, we aim to round a real number down to the nearest whole number.
Whole numbers are vital in both theoretical and practical mathematics as they form the basis for counting and simple arithmetic. They make it possible to perform operations like addition, subtraction, multiplication, and division without the complications that fractions or decimals bring.
Real Number
Real numbers encompass all numbers that exist on the number line, including both rational and irrational numbers. They represent all possible values of quantities that can exist in the real world, encompassing integers, fractions, and non-repeating, non-ending decimals.
Real numbers are divided into several categories:
Real numbers are divided into several categories:
- **Integers:** Whole numbers and their negatives.
- **Rational numbers:** Numbers that can be expressed as the quotient of two integers.
- **Irrational numbers:** Numbers that cannot be written as a simple fraction.