Is the following function even, odd, or neither \( f(x)=8 x^{6}-4 x^{2}-5 \) Neither Even Odd

To determine whether the function \( f(x)=8 x^{6}-4 x^{2}-5 \) is even, odd, or neither, we check the properties of even and odd functions. An even function satisfies \( f(-x) = f(x) \), while an odd function satisfies \( f(-x) = -f(x) \). Calculating \( f(-x) \): \[ f(-x) = 8(-x)^6 - 4(-x)^2 - 5 = 8x^6 - 4x^2 - 5 \] Notice that \( f(-x) = f(x) \). Since this holds true, we can conclude that the function is even. Additionally, understanding the geometric interpretation can enhance your grasp. Even functions are symmetrical about the y-axis, which means if you were to fold the graph along the y-axis, both sides would align perfectly. Picture it as a mirrored reflection of the function on either side of the y-axis. Lastly, for a high-level application of even functions, they frequently appear in physics and engineering, particularly in systems with symmetry. For example, the analysis of structures or waves often utilizes even functions, leading to models that easily lend themselves to predictions about stability or amplitude. This makes them quite handy beyond just schoolwork!