Determine if the function is odd, even, or neither. \( f(x)=3 x^{4}+3\left|x^{3}\right|-3 \)

To determine the nature of the function \( f(x) = 3x^4 + 3|x^3| - 3 \), we first evaluate \( f(-x) \): \[ f(-x) = 3(-x)^4 + 3|(-x)^3| - 3 = 3x^4 + 3| -x^3| - 3 = 3x^4 + 3|x^3| - 3 \] Since \( f(-x) = f(x) \), the function is even. Now let's dip into a little history: The concepts of even and odd functions can be traced back to the advent of algebra in ancient civilizations. Mathematicians from Babylonia, Egypt, and later Greek scholars like Euclid explored symmetry in their geometric constructs, which laid the groundwork for these function classifications. And for some real-world application, identifying whether a function is odd or even can simplify problems in physics, particularly in mechanics and wave phenomena. For instance, knowing that a wave function is even may allow for simplifications in calculations regarding energy distribution, making it easier to predict behaviors in systems like vibrating strings or sound waves!