graph y=1/2cscx and find the domain and range

Answer 1

To start, you need to know what the cosecant graph is. Cosecant is similar to sine in that the extrema of the graphs (the parts where the line turns around) are the same point. What the extrema is for csc (minimum or maximum) is the opposite of what it is for sin, so where sin has a maximum, csc has a minimum. From this, you can know csc must have a discontinuity (not exist) at certain points, and for csc those points are at any point where the x value is a*π, where a is an integer. At those points, the line goes from approaching positive infinity to negative infinity or the other way around.

From this, you can know for a graph csc(x), the range is (-∞,-1]υ[1,∞) and the domain is(-∞,∞)∩(aπ), where a is any integer. There are a few ways of representing the domain in this kind of situation, but that is my way I was taught. Usually, you will be asked for the domain over a set area to avoid that kind of problem.

Now to account for the 1/2, you can treat this as a scaling value. All values for the y axis from before will be scaled by 1/2. This means if the graph was at a y value of 1, it is now 0.5. If it was at a y value of 4, it is now 2. This does nothing to change the domain as it does not affect the period of the function, but it does affect the amplitude so it will affect the range. Taking all the values and applying our scaler to the range, you get (-∞,-0.5]υ[0.5,∞).

Thus, the range for 1/2csc(x) is (-∞,-0.5]υ[0.5,∞), and the domain with my version of formatting is (-∞,∞)∩(aπ), where a is any integer.