Final answer:
The function y = cosx/x is an odd function, as it satisfies the condition y(x) = -y(-x) when x is replaced by -x in the function.
Step-by-step explanation:
To determine whether the function y = cosx/x is even, odd, or neither, we need to analyze its symmetry properties with respect to the y-axis. An even function is symmetric about the y-axis and satisfies the condition y(x) = y(-x), which means that the function is unchanged when x is replaced with -x. On the other hand, an odd function satisfies the condition y(x) = -y(-x) and has symmetry such that if the function is reflected about the y-axis and then about the x-axis, the function looks the same.
Let's test the given function for evenness or oddness. We replace x with -x in the function and get y(-x) = cos(-x)/(-x). Now, since cos(-x) is equal to cos(x) because cosine is an even function, and -x is the negative of x, we have y(-x) = cos(x)/(-x). This simplifies to -cos(x)/x, which is -y(x). Hence, our function satisfies the condition for an odd function. Additionally, we know that the integral over all space of an odd function is zero, adding further support to the conclusion.