Determine whether each function is even, odd, or neither. See Example 5. 1 ƒ(x) = x + —— x⁵

Verified step by step guidance

1. Recall the definitions: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \), and odd if \( f(-x) = -f(x) \) for all \( x \). If neither condition holds, the function is neither even nor odd.
2. Given the function \( f(x) = x + \frac{1}{x^5} \), substitute \( -x \) into the function to find \( f(-x) \): \[ f(-x) = (-x) + \frac{1}{(-x)^5} \].
3. Simplify \( f(-x) \): \[ f(-x) = -x + \frac{1}{-x^5} = -x - \frac{1}{x^5} \].
4. Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \): \[ f(x) = x + \frac{1}{x^5} \quad \text{and} \quad -f(x) = -x - \frac{1}{x^5} \]. Notice that \( f(-x) = -f(x) \).
5. Since \( f(-x) = -f(x) \), conclude that the function \( f(x) = x + \frac{1}{x^5} \) is an odd function.

Key Concepts

Even and Odd Functions

An even function satisfies f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. An odd function satisfies f(-x) = -f(x), indicating symmetry about the origin. Determining whether a function is even, odd, or neither involves testing these conditions.

Function Substitution and Simplification

To test if a function is even or odd, substitute -x into the function and simplify the expression. This process helps compare f(-x) with f(x) and -f(x) to check for symmetry properties. Accurate algebraic manipulation is essential for correct classification.

Properties of Polynomial and Rational Functions

Understanding how powers of x behave under negation is crucial; even powers of x remain positive when x is replaced by -x, while odd powers change sign. For rational functions, consider numerator and denominator separately to determine overall parity.