/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 72 The period of the function $$ ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 72

The period of the function $$ f(x)=\left\\{\begin{array}{ll} 1, & \text { when } x \text { is a rational } \\ 0, & \text { when } x \text { is irrational } \end{array}\right. \text { is } $$ (A) 1 (B) 2 (C) non-periodic (D) None of these

Short Answer

Expert verified
The function is non-periodic.

Step by step solution

01

Understand the function

The function given is a characteristic function that takes the value 1 for rational numbers and 0 for irrational numbers. This type of function does not change periodically between different values in a predictable and regular manner as typically periodic functions do.
02

Define periodicity in context

A function is periodic with period \( T \) if \( f(x + T) = f(x) \) for all \( x \) in the domain. This means that the function repeats itself after every interval of length \( T \).
03

Check for periodicity for rational inputs

Substituting any rational number \( x \), the function returns 1, i.e., \( f(x) = 1 \). For a function to be periodic with non-zero period \( T \), there must exist a \( T \) such that for every rational \( x \), \( x + T \) must also map to a rational number. However, its irrationality distribution doesn't affect with a constant interval; hence no such \( T \) exists.
04

Check for periodicity for irrational inputs

Substituting any irrational number \( x \), the function returns 0, i.e., \( f(x) = 0 \). For a function to have a period \( T \), the addition of \( T \) to any irrational \( x \) should remain consistently 0. Since numerals and interval relations are independent of such consistency, no period satisfies this condition either.
05

Conclusion on periodicity

Since no fixed \( T \) exists that causes \( f(x + T) = f(x) \) for all \( x \), the function cannot be periodic. There isn't a single repeated pattern shared by all rational and irrational outcomes of the function over any interval.

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Key Concepts

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

Rational Numbers
Rational numbers are numbers that can be expressed as a fraction or quotient of two integers, where the numerator is an integer and the denominator is a non-zero integer. This characteristic makes them appear as finite decimals or recurring decimals. For example, the number \( \frac{2}{3} \) is a rational number, as it can be written in fraction form. Other examples include 0.5, which is equivalent to \( \frac{1}{2} \), and the repeating decimal 0.333..., which is equivalent to \( \frac{1}{3} \). Rational numbers are important in mathematics because:
  • They fill in the gaps between integers on the number line.
  • Every integer is a rational number, since any integer \( a \) can be written as \( \frac{a}{1} \).
  • They help us understand quantities and ratios in practical life, like sharing an apple among friends.
Understanding rational numbers is crucial because they help form the foundation for more complex numbers.
Irrational Numbers
Irrational numbers are numbers that cannot be expressed as a fraction of two integers, which means they have non-repeating, non-terminating decimal expansions. Examples include \( \sqrt{2} \), \( \pi \), and \( e \). These numbers cannot be precisely written as fractions, hence the term 'irrational'. The significance of irrational numbers lies in several aspects:
  • They complete the real number line, providing numbers that fill spaces that rational numbers can't.
  • Irrational numbers are essential in geometry, often appearing in calculations involving circles and right triangles.
  • They arise naturally in many mathematical constants and complex equations.
When dealing with irrational numbers, it is important to recognize their unpredictability compared to the orderly nature of rational numbers.
Characteristic Function
A characteristic function, in mathematics, is often used to indicate membership in a set by assigning a specific value to elements inside the set and another value to elements outside the set. In the context of the given exercise, the characteristic function assigns a value of 1 for rational numbers and 0 for irrational numbers. This type of function is a simple yet powerful tool for separating elements based on specific criteria.
Key points about characteristic functions include:
  • They map input values uniquely to a set of predefined outputs.
  • In calculus and analysis, they are beneficial for defining subsets of a given domain clearly.
  • Characteristic functions provide a binary output, making them easy to understand and work with.
In summary, characteristic functions help classify numbers or elements efficiently, based on whether they meet certain conditions.

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Most popular questions from this chapter

Let \(f(x)=\sin ^{-1}(\log [x])+\log \left(\sin ^{-1}[x]\right)\), where [ ] denotes the greatest integer function. Then, (A) domain of \(f\) is \([1,2)\) (B) domain of \(f\) is \([1,3)\) (C) range of \(f\) is \(\left\\{\log \frac{\pi}{2}\right\\}\) (D) range of \(f\) is \(\\{0\\}\)

Let \(f: R \rightarrow R\) be a function defined by, \(f(x)=\) \(-\frac{|x|^{3}+|x|}{1+x^{2}}\), then the graph of \(f(x)\) lies in which quadrant \((s) ?\) (A) I and II (B) I and III (C) II and III (D) III and IV

If the domain for \(y=f(x)\) is \([-3,2]\), then the domain of \(g(x)=f\\{|[x]|\\}\) is (A) \((-2,3)\) (B) \([-2,3]\) (C) \([-2,3)\) (D) \((-2,3]\)

The range of the function \(y=\frac{x-[x]}{1-[x]+x}\) (A) \(\left(0, \frac{1}{2}\right)\) (B) \(\left[0, \frac{1}{2}\right]\) (C) \(\left[0, \frac{1}{2}\right)\) (D) \(\left(0, \frac{1}{2}\right]\)

The domain of the function \(f(x)=\ln \left\\{\operatorname{sgn}\left(9-x^{2}\right)\right\\}+\sqrt{[x]^{3}-4[x]}\), where \([\cdot]\) denotes integral part, is (A) \((-2,1) \cup(2,3)\) (B) \([-2,1) \cup[2,3)\) (C) \([-2,1] \cup[2,3)\) (D) \([-2,1) \cup[2,3]\)
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