Question

POLYNOMIAL FUNCTIONS

ALGEBRA 2

Which statement about the function f(x) = x^3 is true?
A) The function is positive where x < 0.
B) The function never increases.
C) The function is odd.
D) None of the above.

Answer 1

The correct answer to the question about the function f(x) = x^3 is that the function is an odd function, meaning it is symmetric with respect to the origin, and thus is negative for x < 0, and not always increasing or positive for x < 0.

Step-by-step explanation:

The question asks which statement about the function f(x) = x^3 is true. Evaluating the options, we know that:

• A cubic function with the form f(x) = x^n, where n is an odd number, is always an odd function.
• An odd function is symmetric with respect to the origin, meaning that f(-x) = -f(x). Thus, the function will be negative where x < 0.
• An odd function, like a cubic function, generally has parts where it increases and decreases, depending on the range of x values we look at.

Therefore, the correct answer is C) The function is odd, as option A is incorrect because f(x) would be negative where x < 0, option B is incorrect because the function does increase and decrease, and option D can't be correct since we have identified C as correct.