Problem 61 - Calculus

Question

Explain what is wrong with the statement. The function \(-\log |x|\) is odd.

Step-by-Step Solution

Answer

The function is not odd; it doesn't satisfy \(f(-x) = -f(x)\).

Step 1: Understand the Property of Odd Functions

A function \(f(x)\) is considered odd if for all \(x\), the equation \(f(-x) = -f(x)\) holds. This means that the function is symmetric around the origin.

Step 2: Analyze the Given Function

The given function is \(f(x) = -\log |x|\). We need to investigate whether it satisfies the condition of an odd function.

Step 3: Apply the Odd Function Test

To determine if \(f(x) = -\log |x|\) is odd, calculate \(f(-x)\):1. \(f(-x) = -\log |-x| = -\log |x|\).Compare \(f(-x)\) with \(-f(x)\): \(-f(x) = -(-\log |x|) = \log |x|\).

Step 4: Compare the Two Expressions

From Step 3, we found that:- \(f(-x) = -\log |x|\)- \(-f(x) = \log |x|\)Since \(f(-x) eq -f(x)\), the function \(-\log |x|\) does not satisfy the condition for being an odd function.

Step 5: Conclusion

Because the condition \(f(-x) = -f(x)\) is not met, the statement that the function \(-\log |x|\) is odd is incorrect.

Key Concepts

Function Symmetry

Function symmetry revolves around the notion of how functions behave in relation to the y-axis or origin on a graph. A function displays symmetry if it maintains a specific consistent form across these axes. Specifically, we call a function **odd** if it fulfills a particular condition about its values at positive and negative points. This condition is expressed mathematically as:\[ f(-x) = -f(x) \]This equation indicates that for every value of \(x\), the function at \(-x\) is essentially the negative of the function at \(x\). This property causes the graph of odd functions to be symmetric about the origin, forming a rotational symmetry of 180 degrees.

Some common odd functions include \(f(x) = x^3\) and \(f(x) = \sin(x)\), which visibly demonstrate this symmetry. Understanding function symmetry is crucial because it helps in identifying different types of functions quickly and predicting their behavior without needing to graph them extensively.

Logarithmic Functions

Logarithmic functions are unique mathematical expressions with a base and a variable inside a logarithm. They are the inverse of exponential functions, which makes them useful in various domains such as science and engineering. The general form of a logarithm is:\[ \log_b(x) \]where \(b\) is the base and \(x\) is the argument, meaning the number you are taking the log of. The base is usually a number like 10, e (approximately 2.718), or other positive numbers. A special property of logarithms is that multiplication and division within the argument can be expressed as addition and subtraction of logarithms, like:

• \(\log_b(xy) = \log_b(x) + \log_b(y)\)
• \(\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)\)

Logarithms simplify complex multiplications and divisions, turning them into easier addition and subtraction problems which are easier to handle analytically. Despite their seemingly unusual form, they are heavily utilized in solving exponential equations and in various calculations due to their ability to condense large ranges of values.

Mathematical Proofs

Mathematical proofs are a backbone of verifying and understanding mathematical concepts. They play a vital role in confirming whether statements are universally true under defined conditions. Proofs use logical reasoning to demonstrate the validity of a proposition, starting from axioms or already proven statements. Two common types of proof are:

• Direct Proof: Starting from known facts, applying logical steps to arrive at the statement in question.
• Proof by Contradiction: Assuming the opposite of what you want to prove, showing that this assumption leads to a contradiction, thereby confirming the truth of the original statement.

In the context of determining if a function is odd, we rely on proving the functional condition \(f(-x) = -f(x)\). This involves substituting \(-x\) into the function and comparing it with the negative of the function's original form, as shown in the step-by-step analysis of \(f(x) = -\log |x|\). This approach of using mathematical logic and reasoning helps uncover the truth behind various mathematical properties and ensures rigorous validation.