Problem 48 - Thomas Calculus

Question

In Exercises \(47-58,\) say whether the function is even, odd, or neither. Give reasons for your answer. $$ f(x)=x^{-5} $$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Review definitions

A function is called **even** if for every \( x \), \( f(x) = f(-x) \). A function is called **odd** if for every \( x \), \( f(-x) = -f(x) \). If neither of these conditions are satisfied, the function is **neither even nor odd**.

Step 2: Substitute \( -x \) into the function

Replace \( x \) with \( -x \) in the function \( f(x) = x^{-5} \). This gives us: \[ f(-x) = (-x)^{-5} = -x^{-5} \]

Step 3: Compare \( f(-x) \) with \( f(x) \) and \(-f(x)\)

We have \( f(-x) = -x^{-5} \) and \( f(x) = x^{-5} \). Compare these to check if \( f(x) = f(-x) \) or \( f(-x) = -f(x) \): - \( f(x) = x^{-5} \) is not equal to \( f(-x) = -x^{-5} \), so the function is not even.- Check if \( f(-x) = -f(x) \): \(-f(x) = -(x^{-5}) = -x^{-5}\). Hence, \( f(-x) = -f(x) \).

Step 4: Conclusion

Since \( f(-x) = -f(x) \), by definition, the function \( f(x) = x^{-5} \) is odd.

Key Concepts

Function Symmetry

Understanding function symmetry helps us determine if a function is even, odd, or neither. This is especially useful when analyzing polynomial functions or any type of mathematical functions. Functions can be fascinating in how they visually or mathematically mirror certain properties. To check symmetry:

• Even Function: A function is even if for every input, \(x\), the function satisfies \(f(x) = f(-x)\). Geometrically, an even function is symmetric across the y-axis.
• Odd Function: A function is odd if for every input, \(x\), the function satisfies \(f(-x) = -f(x)\). Graphically, it can be thought of as having a sort of rotational symmetry about the origin, as if the graph can be rotated 180 degrees around the origin and still look the same.
• If a function does not satisfy either of these conditions, it is regarded as neither even nor odd.

Checking whether a function is even, odd, or neither helps simplify mathematical analysis and problem-solving, offering insights into the function's inherent properties.

Mathematical Proofs

Mathematical proofs are crucial in verifying concepts like function symmetry by providing a logical, step-by-step demonstration that shows the underlying truth of a statement. To decide whether our function \( f(x) = x^{-5} \) is even, odd, or neither, we apply a concise mathematical proof by substitution and comparison.Here are the steps involved:

• Step 1: Identify your function and write down the symmetry properties to check. For example, \( f(x) = f(-x)\) for even and \( f(-x) = -f(x)\) for odd.
• Step 2: Substitute \( -x \) into the function. In our example, \( f(-x) = (-x)^{-5} \) becomes \(-x^{-5}\).
• Step 3: Compare the result with the original function values. Notice if \( f(x) = f(-x) \) or \( f(-x) = -f(x) \).
• Conclusion: If \( f(-x) = -f(x) \, \ f(x) \) is odd; if \( f(x) = f(-x) \, \ f(x) \) is even. For our specific function, we conclude it is odd.

Mathematical proofs reinforce understanding and validate mathematical properties, ensuring no assumption is merely taken at face value.

Polynomial Functions

Polynomial functions are an essential topic in mathematics, characterized by expressions involving variables raised to whole-number exponents. Each term's contribution adds a layer to the function's overall behavior. Polynomial functions are vital in evaluating symmetry due to structured patterns that can predictably convey symmetry attributes.Power functions like \( f(x) = x^{-5} \) are specific cases of polynomial expressions. Here are a few insights into polynomial functions:

• Degree and Symmetry: In polynomial terms, the exponents play a significant role in determining symmetry. A function \( f(x) = x^n \, \ n \) is even if \ n \ is even, and odd if \ n \ is odd.
• Term Role: Powers of odd degree, such as \( x^{-5}\), naturally lead to odd function behavior.
• Simplifying Analysis: Recognizing these patterns in polynomial functions can simplify the evaluation of unfamiliar functions and guide expectations about their symmetry.

By examining polynomial functions, we develop a deeper intuition about their behavior and the interdependencies between their symmetries and formula characteristics.