Question

find f(-x) - f(x) for the following function. f(x) = x³ + x + 2 f(-x) - f(x) = \square (simplify your answer. do not factor.)

Explanation:

Step1: Find \( f(-x) \)

Substitute \( -x \) into \( f(x) = x^3 + x + 2 \).
\( f(-x) = (-x)^3 + (-x) + 2 = -x^3 - x + 2 \).

Step2: Compute \( f(-x) - f(x) \)

Substitute \( f(-x) = -x^3 - x + 2 \) and \( f(x) = x^3 + x + 2 \):
\( f(-x) - f(x) = (-x^3 - x + 2) - (x^3 + x + 2) \).
Simplify by distributing the negative sign:
\( = -x^3 - x + 2 - x^3 - x - 2 \).
Combine like terms:
\( (-x^3 - x^3) + (-x - x) + (2 - 2) = -2x^3 - 2x \).

Answer:

\( -2x^3 - 2x \)