Determine whether each function is even, odd, or neither. ƒ(x)=x³-x+9

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 \( f(x) = x^{3} - x + 9 \). Substitute \( -x \) into the function:
3. \[ f(-x) = (-x)^{3} - (-x) + 9 \]
4. Simplify the expression for \( f(-x) \): Calculate each term carefully, remembering that \( (-x)^3 = -x^3 \) and \( -(-x) = +x \).
5. Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \) to determine if the function is even, odd, or neither by checking if \( f(-x) = f(x) \) or \( f(-x) = -f(x) \).

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. Functions that do not meet either condition are neither even nor odd.

Function Evaluation and Substitution

To determine if a function is even or odd, substitute -x into the function and simplify. Comparing f(-x) with f(x) and -f(x) helps identify the function's symmetry properties. This process requires careful algebraic manipulation of the given expression.

Polynomial Functions and Their Symmetry

Polynomial functions can be classified by the parity of their terms: even powers contribute to even symmetry, odd powers to odd symmetry. For example, x³ is odd, x² is even, and constants are even. Understanding this helps quickly assess the overall function's symmetry.