# How do you sketch the graph of f(x)=arctan(2x-3)?

Jul 26, 2018

See graph and details.

#### Explanation:

$y = \arctan \left(2 x - 3\right) \in \left(- \frac{\pi}{2} , \frac{\pi}{2}\right)$

$y \ne$asymptotuc $\pm \frac{\pi}{2}$

Inversely,

$x = \frac{1}{2} \left(\tan y + 3\right) \Rightarrow$ the period in y-directions = $\pi$.

So, one period is $y \in \left(\frac{\pi}{2} , \frac{\pi}{2}\right)$, the range of y..

See graph that is restricted to one period:
graph{(y - arctan(2x-3))(y^2-1/4(pi)^2)=0}

For the piecewise-wholesome inverse

$y = {\left(\tan\right)}^{- 1} \left(2 x - 3\right) = k \pi$ + given $y , k = 0 , \pm 1 , \pm 2 , \pm 3 ,$.

the graph is immediate, using the inverse of the given equation

$x = \frac{1}{2} \left(\tan y + 3\right)$.

See graph.
graph{x-1/2(3 + tany)=0[-20 20 -10 10]}
It is a wrong practice to swap ( x, y ) to ( y, x ) and call

$y = \frac{1}{2} \left(\tan x + 3\right)$

the inverse of the given equation, and rotate the graph for the

given equation, for $y \in \left(- \frac{\pi}{2} , \frac{\pi}{2}\right)$.

In te chosen piece, the graphs of both the given equation and its

wholesome inverse ought to be the same. ..