Question

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

Answer 1

See graph and details.

Explanation:

`y = arctan ( 2 x - 3 ) in ( -pi/2, pi/2 )`

`y ne`asymptotuc `+-pi/2`

Inversely,

`x = 1/2 ( tan y + 3 ) rArr` the period in y-directions = `pi`.

So, one period is `y in ( pi/2, pi/2 )`, 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 = (tan)^(-1) ( 2x - 3 ) = kpi` + given `y, k = 0, +-1, +-2, +-3,`.

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

`x = 1/2 ( tan y + 3 )`.

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 = 1/2 ( tan x + 3 )`

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

given equation, for `y in ( - pi/2, pi/2 )`.

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

wholesome inverse ought to be the same. ..