Final answer:
To tell if a function is even or odd by examining its range, you need to look for y-axis symmetry in even functions and origin symmetry in odd functions. The range itself isn't a direct indicator, but the function's values and their symmetry about the y-axis or origin provide the answer.
Step-by-step explanation:
To determine if a function is even or odd by examining its range, you would need to investigate the symmetry of the function. While the range itself doesn't strictly determine whether a function is even or odd, the behavior of the function values does. An even function is symmetric about the y-axis. This means that for every x in the function's domain, f(x) = f(-x). For instance, the parabola y = x² is an even function because f(2) = 4 and f(-2) = 4, demonstrating symmetry across the y-axis. The range doesn't change as x takes on positive or negative values.
On the other hand, an odd function has origin symmetry, which means that if f(x) is part of the function, then -f(x) will also be part of the function at -x. This implies that for every positive value in the range, there is a corresponding negative value (and vice versa). For example, y = x³ is an odd function as f(2) = 8 and f(-2) = -8, showing the points are reflected about the origin.
In summary, for an even function, the range will be the same for x and -x. For an odd function, if a particular y value is part of the range, then -y will also be in the range, provided x and -x belong to the domain. Without symmetry or these characteristics, the function is neither even nor odd.