Determine whether the function \( f(x)=-3 x^{3}+3 x^{7}+x \) is even, odd or neither. Answer neither odd

To determine the nature of the function \( f(x) = -3x^{3} + 3x^{7} + x \), we can evaluate \( f(-x) \): \[ f(-x) = -3(-x)^{3} + 3(-x)^{7} + (-x) = 3x^{3} - 3x^{7} - x. \] Now, comparing \( f(-x) \) with \( f(x) \): - \( f(x) \) includes terms with odd powers (cubic and seventh) and a linear term. - Since \( f(-x) \) is not equal to \( f(x) \) and is not equal to \( -f(x) \), the function is neither even nor odd. In conclusion, the function \( f(x) \) is neither even nor odd. To get a better grasp of even and odd functions, remember that even functions are symmetric about the y-axis, while odd functions exhibit rotational symmetry about the origin. Investigating those characteristics helps in classifying many familiar functions! For a deeper dive into function types, consider exploring polynomial functions in greater detail. Understanding the behaviors of various polynomials—including quadratic, cubic, and higher degrees—can provide insight into how function characteristics manifest in graphs, leading to applications in calculus and beyond!