Determine whether each function is even, odd, or neither. ƒ(x)=x+1/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 it is odd if \( f(-x) = -f(x) \) for all \( x \). If neither condition holds, the function is neither even nor odd.
2. Given the function \( f(x) = x + \frac{1}{x^5} \), start by finding \( f(-x) \). Substitute \( -x \) into the function:
3. \[ f(-x) = (-x) + \frac{1}{(-x)^5} \]
4. Simplify the expression for \( f(-x) \). Remember that \( (-x)^5 = -x^5 \) because an odd power preserves the negative sign:
5. \[ f(-x) = -x + \frac{1}{-x^5} = -x - \frac{1}{x^5} \]
6. Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \): \( f(x) = x + \frac{1}{x^5} \) and \( -f(x) = -x - \frac{1}{x^5} \). Since \( f(-x) = -f(x) \), the function \( f(x) \) is odd.

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), showing symmetry about the origin. Determining whether a function is even, odd, or neither involves testing these conditions.

Function Substitution

To test if a function is even or odd, substitute -x into the function in place of x. Simplify the resulting expression and compare it to the original function f(x) and its negative -f(x). This substitution is key to identifying symmetry properties.

Properties of Exponents and Algebraic Simplification

Understanding how to manipulate expressions with exponents, especially negative and fractional powers, is essential. Simplifying terms like (−x)^5 or 1/(−x)^5 correctly helps in comparing f(-x) with f(x) or -f(x) accurately.