Question

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

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

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 \( F eq G \).

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.

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.

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 \([0,1)\).

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.

Key concepts

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.