What is the domain and range of #y = tan(2x)#?

1 Answer
Alvin L.
Dec 16, 2017

Domain: #{x|x!=pi/4+pi/2k,k inZZ}#
Range: #y in RR#

Explanation:

Let us first look at the graph of #y=tan(2x)#:
graph{tan(2x) [-8.41, 9.37, -3.74, 5.146]}
We can see that it has recurring vertical asymptotes, which means that the function is undefined at all these points.

To find the asymptotes, we will look at the following identity:
#tan(theta)=sin(theta)/cos(theta)#
#tan(2x)=sin(2x)/cos(2x)#
This equation tells us that the vertical asymptotes occur when #cos(2x)=0#, and this happens when #x=pi/4+pi/2k# where #k inZZ#

Since the function is defined for all but those #x# values, we just need to exclude them from our domain, so we have:
#{x|x!=pi/4+pi/2k,k in ZZ}#

Next we want to look at the range, and we see on the graph that it goes from #-oo# to #oo# (since each "section" is bounded by two vertical asymptotes), and since the function is continuous on that interval, we know that the domain is all real numbers:
#{y inRR}#

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