# How do you find the domain, range, and asymptote for y = 1 - tan ( x/2 - pi/8 )?

Apr 10, 2018

Ask yourself where the function is defined.

#### Explanation:

The function $y = 1 - \tan \left(x \frac{\pi}{2} - \frac{\pi}{8}\right)$ is not defined where $\tan \left(x \frac{\pi}{2} - \frac{\pi}{8}\right)$ is not defined $\implies x \frac{\pi}{2} - \frac{\pi}{8} = k \frac{\pi}{2}$, $k \in \mathbb{Z}$
$\frac{x}{2} - \frac{1}{8} = \frac{k}{2}$
$x = k + \frac{1}{4}$
The domain is RR-{(k+1/4) pi/2}; k in ZZ, the range is $\mathbb{R}$

An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.(from wikipedia)

In this case the asymptotes are vertical lines x=(k+1/4) pi/2; k in ZZ

Some help:
1.
Remember about $\tan \left(x\right)$:
$\tan \left(x\right)$ The domain is RR-{k pi/2}; k in ZZ, the range is $\mathbb{R}$

Look at the unit circle, the distance between the x-ax and the intersection of the green and blue line is $\tan \left(x\right)$, where $x$ is the angle. If $x \rightarrow \frac{\pi}{2}$ there is no intersection of the green and blue line, there $\tan \left(x\right)$ is not defined.

graph{tan(x) [-5, 5, -2.5, 2.5]}

2.
$a \cdot \tan \left(b \cdot x - c\right) + d$
$a$- change the slope of the graph
$b$-change the frequency (the "lines" are more close or away to each other)
$c$-translation parallel to y-ax, move the graph right or left
$d$-translation parallel to x-ax, move the graph up or down
You can use this graphing calculator write in $a \cdot \tan \left(b \cdot x - c\right) + d$ and play with sliders.