Question

How do you find the domain and range of `arctan(x^2)`?

Answer 1

Range: `y = arctan (x^2) in [0, pi/2 )`,
sans the asymptotic `y = pi/2`. .
Domain: `x in ( - oo, oo )`.

Explanation:

`y = arctan x^2 rArr 0`, as `x^2 to 0 rArr x to 0`.

By convention, arctan values are confined to `( -pi/2, pi/2 )`.

Inversely, `x = +- sqrt( tan y), tan y >=0 rArr y in [0, pi/2)`

Here, it is halved, as `x^2 >= 0`. See illustrative graph.

graph{(y-arctan(x^2))(y-pi/2)=0}.

For the interested readers, some related information;

Using the piecewise-wholesome inverse operator (tan)^(-1),

instead of `tan^(-1)`,

`y = (tan)^(-1)(x^2)`

and using its inverse `x^2 = tan y`

the graph that is same for both is created.

graph{x^2- tan y= 0}

The y-negative graphs are constituents of

`y = (tan)^(-1)( x^2 )= kpi + arctan x^2, k = 0, +-1, +-2, +-3, .`