# What is the domain and range of y= 3 tan x?

Dec 16, 2017

Domain: $\left\{x | x \ne \frac{\pi}{2} + \pi k , k \in \mathbb{Z}\right\}$
Range: $y \in \mathbb{R}$

#### Explanation:

graph{3tanx [-10, 10, -5, 5]}
As we can see from the graph, there are recurring vertical asymptotes, and this means that the function is not defined at these points. So we need to find these points and exclude them from our domain.
To do this, we will take help of the $\tan \left(\theta\right) = \sin \frac{\theta}{\cos} \left(\theta\right)$ identity. This means our function will produce a vertical asymptote when $\cos \left(x\right) = 0$, which happens when $x = \frac{\pi}{2} + \pi k$, where $k \in \mathbb{Z}$.

Now we know all the points where our function isn't defined, so we know the domain must be:
$\left\{x | x \ne \frac{\pi}{2} + \pi k , k \in \mathbb{Z}\right\}$

Now for the range. We see that the all of the sections between the vertical asymptotes go from $- \infty$ to $\infty$, so the range is all real numbers:
$y \in \mathbb{R}$