Final answer:
To find the x-intercepts of the function f(x) = sin(4x)/2x, we solve for x when sin(4x) = 0, leading to x-intercepts at x = ⅓nπ for all integers n, excluding n = 0.
Step-by-step explanation:
The student is asking about the x-intercepts of the function f(x) = sin(4x)/2x. To find the x-intercepts of a function, we look for the values of x that make f(x) equal to zero. In the case of the sine function, sin(4x) equals zero when 4x is an integer multiple of π (pi), because the sine of zero and multiples of pi is zero. Therefore, set sin(4x) = 0 and solve for x:
4x = nπ
x = ⅓nπ, where n is an integer
However, since the expression is sin(4x)/2x, we must also consider the 2x in the denominator; the function is undefined at x = 0. Thus, the x-intercepts occur at x = ⅓nπ for all integers n except n=0.