Problem 3 - A Book of Abstract Algebra

Question

Proof By exhibiting a counterexample: \(-1\) is not equal to \(f(x)\) for any \(x \in \mathbb{R}\). \(f(x)=|x|\)

Step-by-Step Solution

Answer

The function \(f(x) = |x|\) cannot be equal to \(-1\) because \(|x|\) is always non-negative.

Step 1: Understanding the Problem

We are asked to show that for the function \(f(x) = |x|\), there is no real number \(x\) such that \(f(x) = -1\). The problem can be approached by considering the properties of the absolute value function.

Step 2: Analyze the Absolute Value Function

The function \(f(x) = |x|\) is the absolute value function. For any real number \(x\), \(|x| \geq 0\). This means the output of \(f(x)\) is always non-negative.

Step 3: Identify a Counterexample

To show that \(-1\) is not a value of \(f(x)\), we need to demonstrate that \(f(x)\) can never be negative. Since \(f(x) = |x|\) outputs values that are zero or positive, there is no \(x\) such that \(f(x) = -1\).

Step 4: Conclusion with Counterexample

The value \(-1\) is outside the range of \(|x|\), which is \([0, \infty)\). Therefore, there is no real number \(x\) for which \(f(x) = -1\). This serves as the counterexample illustrating \(f(x)\) can never equal \(-1\).

Key Concepts

Absolute Value Function

The absolute value function is a simple yet powerful mathematical tool. It is denoted as \(f(x) = |x|\), and it measures the distance of a number from zero on the number line. This means that no matter whether you input a positive number, a negative number, or zero itself, the output will remain a non-negative number.

Here’s a quick breakdown of how the function behaves:

• If \(x > 0\), then \(|x| = x\).
• If \(x < 0\), then \(|x| = -x\), which flips the negative sign to make it positive.
• If \(x = 0\), then \(|x| = 0\).

The function essentially strips away any negativity from the input value, ensuring that the output remains non-negative. This feature is critical in mathematics as it allows for straightforward computations and comparisons that are bound by positive results.

Real Numbers

Real Numbers are the set of numbers that include all the integers, fractions, and decimals that do not involve imaginary numbers. Represented by \(\mathbb{R}\), the real number system is a comprehensive set crucial in mathematical analysis and everyday computations.

Real numbers can be categorized into:

• Natural numbers: Counting numbers starting from 1, 2, 3, and so forth.
• Whole numbers: Natural numbers including zero.
• Integers: Positive and negative whole numbers, including zero.
• Rational numbers: Numbers that can be expressed as a fraction or ratio of two integers.
• Irrational numbers: Numbers that cannot be expressed as a simple fraction, like \(\pi\) or the square root of 2.

The vast range and flexibility of real numbers make them indispensable. In the context of absolute value functions, when we input a real number, the function will always transform it into a non-negative output.

Non-Negative Outputs

When we speak about non-negative outputs, we're referring to values that are either positive or zero. In mathematical functions, some are designed to ensure outputs do not dip below zero, such as the absolute value function. This characteristic is particularly beneficial in various real-world situations where negative values don't make practical sense, such as measuring distance or weighing objects.

A key trait of the absolute value function \(f(x) = |x|\) is that it will

• Only return values that are non-negative.
• Guarantee the minimum value is zero.
• Exhibit a range from zero to positive infinity \( [0,\infty) \).

This means that regardless of the real number plugged into the function, the result will never be negative. It’s impossible for the output to be \(-1\) because the delicate design of the function always keeps the range above or equal to zero. Such properties are crucial for ensuring that the function doesn't violate its intended purpose of producing only non-negative results. Thus, when tasked with proving by counterexample that \(f(x) = -1\) cannot happen, the understanding of non-negative outputs is fundamental.