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Chapter 4: Problem 17

Find two functions \(F, G: \mathbb{R} \rightarrow \mathbb{R}\) where \(F \neq G\) but \(\left.F\right|_{[0,1)}=\left.G\right|_{[0,1)}\)

Short Answer

Expert verified
Define \( F(x) = x^2 \) and \( G(x) = x^2 \) for \(0 \leq x < 1\) and \(G(x) = x\) for \(x \geq 1\).

Step by step solution

01

Understanding the Problem

The task is to find two functions, \( F \) and \( G \), that are defined on the real numbers, such that the restriction of these functions to the interval \([0,1)\) results in the same function, even though \( F eq G \).
02

Define Function F

Define the function \( F(x) \) as \( F(x) = x^2 \) for all real numbers \( x \). This function is continuous and defined for every real number.
03

Define Function G

Define the function \( G(x) \) as follows: \( G(x) = x^2 \) for \( 0 \leq x < 1 \), and \( G(x) = x \) for \( x \geq 1 \). Essentially, \( G(x) \) equals \( F(x) \) on the interval \([0,1)\), but differs beyond that interval.
04

Verify the Restriction Condition

Verify that the restriction of both functions \( F \) and \( G \) over the interval \([0,1)\) is the same. On this interval, \( F(x) = x^2 \) and \( G(x) = x^2 \), so \( \left.F\right|_{[0,1)} = \left.G\right|_{[0,1)} = x^2.\) Thus, they are the same on \([0,1)\).
05

Confirm Functions are Different

To confirm that \( F eq G \), inspect values outside \([0,1)\). For example, at \( x = 2 \), \( F(2) = 4 \) while \( G(2) = 2 \). Since \( F(x) \) and \( G(x) \) produce different results outside \([0,1)\), the functions are different.

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Key Concepts

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

Real-Valued Functions
When we talk about real-valued functions, these are functions that take a set of inputs (in this case, real numbers) and map them to real numbers. This is denoted as: \( f : \mathbb{R} \rightarrow \mathbb{R} \). A common example is the function that squares its input: \( f(x) = x^2 \).
  • These functions are defined on the entire set of real numbers.
  • They map any real input to a real output.
  • They are essential in continuous mathematics and calculus.
Real-valued functions are versatile, allowing for a wide range of domains and codomains, which makes them handy in modeling real-world phenomena, such as calculating areas or predicting patterns.
Function Restriction
Function restriction involves narrowing down the domain of a function to a specific subset. Think of it like taking a large map and zooming into a smaller area. For example, if we have a function \( f(x) = x^2 \), its restriction to the interval \([0,1)\) would only consider the inputs between 0 and 1, not including 1 itself.
  • The notation \( \left.f\right|_{A} \) represents the function \( f \) restricted to the subset \( A \).
  • This is useful for analyzing specific behaviors or characteristics of functions within particular domains.
  • In the given exercise, \( \left.F\right|_{[0,1)} = \left.G\right|_{[0,1)} \) shows how both functions look identical within the specified interval, even if they differ elsewhere.
By focusing on a specific part of the domain, mathematicians can isolate and study particular aspects or properties of a function without distraction from the rest of its behavior.
Continuity in Functions
Continuity in a function means that small changes in the input result in small changes in the output. In technical terms, a function is said to be continuous if it doesn't have abrupt changes or jumps. This can be seen in the squared function, \( f(x) = x^2 \), which is continuous, smooth, and without breaks across all real numbers.
  • Continuity ensures that the graph of a function can be drawn without lifting the pencil.
  • A continuous function on its entire domain appears uninterrupted.
  • In the context of our exercise, both \( F(x) \) and the restricted part of \( G(x) \), \( G(x) = x^2 \) for \([0,1)\), are continuous.
Understanding continuity helps in analyzing and predicting function behavior over intervals and is a crucial concept in calculus, helping in the simplification of complex problems.

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Most popular questions from this chapter

Suppose someone (say, Aesop) is marking days in some leap year (say, 2948). You do not know which days he marks, only how many. Use this to answer the following questions. (Warning: Some, but not all, of these questions use the Pigeon-Hole Principle.) (a) How many days would Aesop have to mark before you can conclude that he marked two days in January? (b) How many days would Aesop have to mark before you can conclude that he marked two days in February? (c) How many days would Aesop have to mark before you can conclude that he marked two days in the same month? (d) How many days would Aesop have to mark before you can conclude that he marked three days in the same month? (e) How many days would Aesop have to mark before you can conclude that he marked three days with the same date (for example, the third of three different months, or the 3 ist of three different months)? (f) How many days would Aesop have to mark before you can conclude that he marked two consecutive days (for example, January 31 and February 1 )? (g) How many days would Aesop have to mark before you can conclude that he marked three consecutive days?

Prove that the function \(F:(0,1) \rightarrow \mathbb{R}\) defined as \(F(x)=(1 / 2-x) /(x(1-x))\) is a biiection.

Prove that every subset of \(\mathbb{N}\) is countable.

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Let \(A\) and \(B\) be nonempty sets, and let \(F: A \rightarrow B\) be a function. Prove that the following are equivalent: (a) \(F\) is \(I-I\). (b) There is a function \(G: B \rightarrow A\) such that \(G \circ F=I d_{A}\). (c) For any set \(C\) and for functions \(H_{1}: C \rightarrow A\) and \(H_{2}: C \rightarrow A,\) if \(F \circ H_{1}=F \circ\) \(\mathrm{H}_{2},\) then \(\mathrm{H}_{\mathrm{l}}=\mathrm{H}_{2}\)
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