Final answer:
The function y = sin(x) / x is neither even nor odd because while sin(x) is an odd function, the presence of the division by x means we cannot cleanly satisfy the conditions for odd or even functions, making the function neither even nor odd.
Step-by-step explanation:
Function Symmetry: Even, Odd, or Neither
When discussing the symmetry of functions in mathematics, it is important to understand what defines even and odd functions. An even function satisfies the condition y(x) = y(-x), meaning that its graph is symmetric about the y-axis. A simple example is x², since (-x)² = x². Conversely, an odd function satisfies y(x) = -y(-x), which results in symmetry about the origin; its graph is symmetric with respect to the y-axis followed by the x-axis. An easy example of an odd function is x sin x, where both x and sin x are odd, and their product remains even.
To determine whether the function y = sin(x) / x is even, odd, or neither, we need to test it against these conditions. We analyze:
• If y(-x) = sin(-x) / (-x) = -sin(x) / x = -y(x), then it is odd.
• If y(-x) = sin(-x) / (-x) = sin(x) / x = y(x), then it is even.
• If neither condition is met, the function is neither even nor odd.
Since sin(x) is inherently an odd function because sin(-x) = -sin(x) and dividing by -x instead of x simply changes the sign, this seems to suggest that y = sin(x) / x may be odd. However, upon closer inspection, because we cannot simplify sin(x) / x to remove the -1 from both numerator and denominator, the function sin(x) / x does not meet the criteria for even or odd symmetry completely, making the function neither even nor odd.