Determine whether each function is even, odd, or neither. ƒ(x)=-x³+2x

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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.

Given the function $ f(x) = -x^{3} + 2x $, find $ f(-x) $ by substituting $ -x $ into the function:

\[ f(-x) = -(-x)^{3} + 2(-x) \]

Simplify the expression for $ f(-x) $ by evaluating the powers and multiplying:

\[ f(-x) = -(-x^{3}) - 2x = x^{3} - 2x \]

Compare $ f(-x) $ with $ f(x) $ and $ -f(x) $ to determine if the function is even, odd, or neither:

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 in place of x and simplify. Comparing the result to the original function f(x) helps identify symmetry properties. This process is essential for analyzing the function's behavior.

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 an odd function term, while x² is even. Understanding this helps predict the overall function's symmetry.

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