Question

Find the domain and range of y = -3tan(3x+pi) +2

Answer 1

For this type of problems, it is best to first graph the "main" function, namely, \(\tan x\):

http://i45.tinypic.com/35cn6m8.png

You can immediately see that the range of the function stretches from \(-\infty\) to \(\infty\). The domain, however, stretches from \(-\infty\) to \(\infty\) as well, but leaves out the points \(k\pi/2\), for all integers \(k\). Because adding \(\pi\) to the function leads to an overlapping, left shift, we have that \(\tan(x)=\tan(x+\pi)\). Hence, the only change to the domain happens when you multiply \(x\) by \(3\): \(\tan(3x)\). This "halves" the frequency of the function's period by \(1/3\).

If normally you have a domain of \(\mathbb{R}\setminus\{k\pi/2\}\), where \(k\in\mathbb{Z}\), can you figure out the new domain if we are now "halving" it by \(1/3\)?

Answer 2

kpi/6?

Answer 3

will I have to remember those limits on the domain and range for the future?

Answer 4

You're correct, and the only thing I'd suggest remembering is the period of the most common trigonometric functions.