Question

determine algebraically whether the function is even, odd, or neither.

(f(x) = 2x^3 - 4)

a. even
b. odd
c. neither

Explanation:

State the algebraic definitions

Using the Even and Odd Functions knowledge point
An even function satisfies \(f(-x) = f(x)\) for all \(x\) in its domain. An odd function satisfies \(f(-x) = -f(x)\) for all \(x\) in its domain. If neither condition is met, the function is classified as neither.

Evaluate the function at negative x

Using the Function Evaluation knowledge point
Substitute \(-x\) into the given function \(f(x) = 2x^3 - 4\):

$$
LATEXBLOCK0
$$

Compare with the original function

Using the Even and Odd Functions knowledge point

$$
LATEXBLOCK1
$$

Thus, the function is not even.

Compare with the negative of the function

Using the Even and Odd Functions knowledge point

$$
LATEXBLOCK2
$$

Thus, the function is not odd.

Determine the final classification

Using the Even and Odd Functions knowledge point
Since \(f(-x)
eq f(x)\) and \(f(-x)
eq -f(x)\), the function is neither even nor odd.

Answer:

• A. even
• B. odd
• C. neither (Correct answer)