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The function y= sin x/x is ____ function, and the function y= cos x/x is ____ function.

User Ddreian
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Final answer:

The function y = sin x/x is an odd function, while the function y = cos x/x is not strictly even or odd due to its discontinuity at x = 0. Even and odd functions are identified by their symmetries, but the presence of x in the denominator modifies these properties for y = cos x/x.

Step-by-step explanation:

The function y= sin x/x is an odd function, and the function y= cos x/x is an even function. To determine the parity of these functions, we can use the definition of even and odd functions.

An even function is symmetric about the y-axis, meaning that f(x) = f(-x) for all x in the function's domain. Conversely, an odd function exhibits symmetry with respect to the origin, which implies that -f(x) = f(-x).

Considering the function y = sin x / x, we notice that sine is an odd function because sin(-x) = -sin(x), and the division by x, which is also an odd function, maintains the oddness of the entire expression. Therefore, -y(x) = -sin(x)/x = y(-x), satisfying the condition for an odd function.

On the other hand, the function y = cos x / x involves the cosine function, which is even because cos(-x) = cos(x), and again, division by x, an odd function. Since an even function divided by an odd function yields an odd function, this might suggest that cos x / x is odd. However, due to the cosine function's even symmetry and the x in the denominator introducing an asymmetry, the overall function does not satisfy the properties of odd functions entirely nor even functions perfectly due to the discontinuity at x = 0. This results in a function that is neither even nor odd. Yet, for x ≠ 0, it often superficially resembles even symmetry because cos(-x) = cos(x), but the discontinuity means we cannot definitively classify it as even.

It is worth noting that the presence of an asymptote in functions like y = 1/x, as mentioned for y = cos x/x and y = sin x/x, suggests that neither x nor y can be zero, which aligns with the fact that both cos x/x and sin x/x are undefined at x = 0.

User Marky
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