Final answer:
Option B, f(x) = 4x^5 + 3x^3 - 2, is neither symmetric about the y-axis nor has order 2 rotational symmetry about the origin because it doesn't satisfy the properties of even or odd functions due to the constant term.
Step-by-step explanation:
To determine which of the given functions is neither symmetric about the y-axis nor has order 2 rotational symmetry about the origin, we need to understand even and odd functions. A function f(x) is even if f(x) = f(-x) and is symmetric about the y-axis; it is odd if f(x) = -f(-x) and has rotational symmetry of order 2 about the origin.
Let's analyze the given options:
- A. f(x) = 6x² - x: This function is neither even nor odd since 6x² is even and -x is odd; however, the combination of both does not satisfy neither the even nor the odd function properties.
- B. f(x) = 4x⁵ + 3x³ - 2: This function is neither even nor odd since the power 5 and 3 terms are odd, but the constant term -2 makes it neither symmetric about the y-axis nor rotationally symmetric about the origin.
- C. f(x) = -x³ + 3x: This function is odd because it satisfies the condition f(x) = -f(-x).
- D. f(x) = x: This function is also odd for the same reason as above.
Therefore, the correct answer to the question is B. f(x) = 4x⁵ + 3x³ - 2, as this function is neither symmetric about the y-axis nor has rotational symmetry about the origin.