Is the function f(x) = cscx continuous if we restrict the domain to the closed interval [-3pi, 3pi] (in radians)? If not, where are the discontinuities?
Ok, so this is how I solve a problem like this:
cscx = 1/sinx
I am imagining a unit circle right now.
At x = 0 radians, f(x) is undefined (because 1/sin0 is the same as 1/0, which is undefined). So already we can answer the first question, and say that the function is not continuous over the set of real numbers contained within the restricted domain.
But to answer the second question:
The only discontinuities that occur are at the points where sinx = 0 (because 1/sinx is undefined when sinx = 0).
Within the closed interval [-3pi, 3pi], sinx = 0 when x = -3pi, -2pi, -pi, 0, pi, 2pi, and 3pi.
Therefore, the discontinuities occur at the points where x = -3pi, -2pi, -pi, 0, pi, 2pi, and 3pi.
I am also supposed to draw a sketch of the graph according to the question I'm working on, but there's no way I'm going to waste time explaining here how I figured out the right way to sketch the graph (it would take too much time to explain in "static" words). Suffice to say, according to Desmos, I got it right, so the answers above should be correct.
Ok, so this is how I solve a problem like this:
cscx = 1/sinx
I am imagining a unit circle right now.
At x = 0 radians, f(x) is undefined (because 1/sin0 is the same as 1/0, which is undefined). So already we can answer the first question, and say that the function is not continuous over the set of real numbers contained within the restricted domain.
But to answer the second question:
The only discontinuities that occur are at the points where sinx = 0 (because 1/sinx is undefined when sinx = 0).
Within the closed interval [-3pi, 3pi], sinx = 0 when x = -3pi, -2pi, -pi, 0, pi, 2pi, and 3pi.
Therefore, the discontinuities occur at the points where x = -3pi, -2pi, -pi, 0, pi, 2pi, and 3pi.
I am also supposed to draw a sketch of the graph according to the question I'm working on, but there's no way I'm going to waste time explaining here how I figured out the right way to sketch the graph (it would take too much time to explain in "static" words). Suffice to say, according to Desmos, I got it right, so the answers above should be correct.
