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

Dec 16, 2017

Domain: $\left\{x | x \ne \frac{\pi}{4} + \frac{\pi}{2} k , k \in \mathbb{Z}\right\}$
Range: $y \in \mathbb{R}$

#### Explanation:

Let us first look at the graph of $y = \tan \left(2 x\right)$:
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:
$\tan \left(\theta\right) = \sin \frac{\theta}{\cos} \left(\theta\right)$
$\tan \left(2 x\right) = \sin \frac{2 x}{\cos} \left(2 x\right)$
This equation tells us that the vertical asymptotes occur when $\cos \left(2 x\right) = 0$, and this happens when $x = \frac{\pi}{4} + \frac{\pi}{2} k$ where $k \in \mathbb{Z}$

Since the function is defined for all but those $x$ values, we just need to exclude them from our domain, so we have:
$\left\{x | x \ne \frac{\pi}{4} + \frac{\pi}{2} k , k \in \mathbb{Z}\right\}$

Next we want to look at the range, and we see on the graph that it goes from $- \infty$ to $\infty$ (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:
$\left\{y \in \mathbb{R}\right\}$