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

Question

Avery Alexander · Accepted Answer

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

#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. y-arctan(x^2))(y-pi/2)=0} is the graph. Here are some related details for those who are interested: Utilizing the inverse operator (tan)^(-1), which is piecewise-wholesome, instead of #tan^(-1)#, #y = (tan)^(-1)(x^2)# and using its inverse #x^2 = tan y# A graph that is identical to both is produced. graph{tan y= 0},x^2- The graphs with y-negative values are components of #y = (tan)^(-1)( x^2 )= kpi + arctan x^2, k = 0, +-1, +-2, +-3, .#

Kennedy Adams · Answer

undefined To find the domain and range of arctan(x^2), the domain is all real numbers since the input (x^2) can take any real value. The range of arctan(x^2) is (-π/2, π/2) because the range of arctan function is (-π/2, π/2), and the square of any real number is non-negative, resulting in positive values only.