Determine whether each function is even, odd, or neither. See Example 5. ƒ(x)=x⁵-2x³

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. Start by finding \( f(-x) \) for the given function \( f(x) = x^5 - 2x^3 \). Substitute \( -x \) into the function: \( f(-x) = (-x)^5 - 2(-x)^3 \).
3. Simplify the powers: \( (-x)^5 = -x^5 \) because an odd power preserves the negative sign, and \( (-x)^3 = -x^3 \) for the same reason. So, \( f(-x) = -x^5 - 2(-x^3) \).
4. Simplify further: \( f(-x) = -x^5 + 2x^3 \).
5. Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \): \( f(x) = x^5 - 2x^3 \) and \( -f(x) = -x^5 + 2x^3 \). Since \( f(-x) = -f(x) \), the function 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), indicating 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 essential for verifying the symmetry properties of the function.

Polynomial Functions and Their Symmetry

Polynomial functions can be analyzed term-by-term for evenness or oddness: terms with even powers of x are even functions, and terms with odd powers are odd functions. The overall function's parity depends on the combination of these terms, which helps in quickly assessing the function's symmetry.