# How do you find the domain & range for (1/2)tan(x)?

##### 1 Answer
Oct 16, 2015

Domain: $\setminus m a t h \boldsymbol{R} \setminus \setminus \left\{\frac{\pi}{2} + k \pi , \setminus k \setminus \in \setminus m a t h \boldsymbol{Z}\right\}$

Range: $\left(- \infty , \infty\right)$.

#### Explanation:

The range is the same of the tangent, because if it is legit to compute $\tan \left(x\right)$, then dividing that value by $2$ is surely ok.

So, the domain is the real set minus the points in which $\cos \left(x\right) = 0$, i.e.

$\setminus m a t h \boldsymbol{R} \setminus \setminus \left\{\frac{\pi}{2} + k \pi , \setminus k \setminus \in \setminus m a t h \boldsymbol{Z}\right\}$

The same goes for the range: since the tangent ranges from $- \infty$ to $\infty$, half of this range is still $\left(- \infty , \infty\right)$.

Note that this is true only because the interval is infinite in both direction, if the original range of $f \left(x\right)$ is $\left[a , b\right]$, then the range of $f \frac{x}{2}$ is $\left[\frac{a}{2} , \frac{b}{2}\right]$.