17) Given the function \( f(x)=x^{3}-3 x \), which of the following is true? \( \begin{array}{ll}\text { A) } f(3)+f(3)=f(6) & \text { C) } f(-3)=f(3) \\ \text { B) } f(-3)=-f(3) & \text { D) } 3 f(3)=f(9)\end{array} \)

To solve this problem, let's explore the function \( f(x) = x^3 - 3x \). First, we calculate \( f(3) \) and \( f(-3) \). 1. \( f(3) = 3^3 - 3(3) = 27 - 9 = 18 \) 2. \( f(-3) = (-3)^3 - 3(-3) = -27 + 9 = -18 \) Now we can evaluate the statements provided: - A) \( f(3) + f(3) = f(6) \) - \( 18 + 18 = f(6) \) → \( 36 \neq 6^3 - 3(6) = 216 - 18 = 198 \) → Incorrect. - B) \( f(-3) = -f(3) \) - \( -18 = -18 \) → Correct. - C) \( f(-3) = f(3) \) - \( -18 \neq 18 \) → Incorrect. - D) \( 3f(3) = f(9) \) - \( 3 \cdot 18 = 54 \) and \( f(9) = 9^3 - 3(9) = 729 - 27 = 702 \) → Incorrect. Thus, the true statement is B: \( f(-3) = -f(3) \). Now for some fun facts! The function \( f(x) = x^3 - 3x \) has a deep connection with calculus, particularly in the study of stationary points and inflections. It becomes truly fascinating when you realize it's rooted in the field of symmetries! If you're itching to dive deeper into the world of polynomial functions, I recommend checking out "Algebra and Trigonometry" by Michael Sullivan. This book really lays down the foundational concepts you'll love exploring!