Is the function \( f(x)=5 x^{5}+7 x+8 \) even, odd, or neither?

To determine if the function \( f(x) = 5x^5 + 7x + 8 \) is even, odd, or neither, we can check the properties of even and odd functions. A function is even if \( f(-x) = f(x) \) and odd if \( f(-x) = -f(x) \). Calculating \( f(-x) \): \[ f(-x) = 5(-x)^5 + 7(-x) + 8 = -5x^5 - 7x + 8 \] Now, compare \( f(-x) \) to \( f(x) \) and \( -f(x) \): 1. \( f(-x) \) does not equal \( f(x) \), so it is not even. 2. Comparing \( f(-x) \) to \( -f(x) \): \[ -f(x) = - (5x^5 + 7x + 8) = -5x^5 - 7x - 8 \] Since \( f(-x) \neq -f(x) \) (the constant term \( +8 \) doesn't match \( -8 \)), the function is not odd either. Thus, \( f(x) \) is neither even nor odd. To understand how functions are categorized as even or odd, it often helps to visualize their graphs. An even function will have symmetrical properties across the y-axis, while odd functions will have rotational symmetry around the origin. This distinction could make for some interesting patterns when plotting different polynomial functions! Also, keep in mind common mistakes like forgetting to apply the negative sign correctly or misanalyzing the constants, which can easily lead to incorrect conclusions about the function's properties. Always double-check your calculations!