Final answer:
To determine if a function is odd or even, you must analyze the function's symmetry with respect to the y-axis or origin. Even functions have y-axis symmetry (y(x) = y(-x)), while odd functions are symmetric about the origin (y(x) = -y(-x)). The correct method is option c) By analyzing the function's symmetry.
Step-by-step explanation:
To determine if a function is odd or even, the key characteristic to look at is the function's symmetry with respect to the origin or the y-axis. This is described in option c) By analyzing the function's symmetry.
An even function satisfies the condition y(x) = y(-x), meaning that if you were to reflect the function across the y-axis, it would coincide perfectly with its original shape. Conversely, an odd function satisfies the condition y(x) = -y(-x), indicating that the function is symmetric about the origin. This means that if you reflect the function across the y-axis and then the x-axis, it too will coincide with its original shape.
It is not correct to evaluate a function's even or odd nature simply by evaluating it at x=0, checking the function's end behavior, or finding its maximum value as suggested in options a), b), and d). For instance, the function xe-x² (an odd function) or x²e-x² (an even function) exemplify that only the symmetry properties can definitively classify functions as odd or even. The correct option in this context is option c, By analyzing the function's symmetry.