Question

f(x) = sin (4x)/2x what are the x intercepts between 0 and pi..

Answer 1

f(x) = sin(4x)/2x

To find x-intercepts, we let y = 0.

0 = sin(4x)/2x, x not= 0, because if it was then the denominator would be 0 which we can't have. Since we are looking for x-intercepts when the function is 0, we can ignore the denominator because we know that if the numerator is 0, then the whole fraction 0. And we have already stated that x not= 0.

Assuming that you are looking for zeroes in the interval [0, pi], we can ignore a zero at x = 0 because it makes our denominator 0. So we look for any other zeroes in that interval. To do so, we first find the period of the function.

Period = 2pi / (coefficient of x) --> Here the coefficient of x is 4. So:
Period = 2pi / 4
= pi/2
So the function repeats once every pi/2 or twice in the interval [0, pi].

Now we let the numerator equal 0 to find the x-intercepts, because when numerator is 0, the whole fraction is 0, and that's where we get our x-ints.

sin(4x) = 0, [0, pi]
If we graph y= sin(4x), from our period, we know that the function repeats every pi/2 meaning it will look like two sin(x) graphs squished in to the [0, pi] interval. We will notice that sin(4x) = 0 when x = 0, pi/4, pi/2, 3pi/4, pi. We remove 0 from our possible 0s since we know that if x = 0, then our denominator becomes 0. Eliminating 0 leaves us with: x = pi/4, pi/2, 3pi/4, pi, which are our x-intercepts for y=sin(4x)/2x in [0, pi].

Answer 2

@mandarin