Question

what is the range of definition in y=arctg(2x+1) ? please with explanation

Answer 1

arctan(x) = tan(x)^(-1), which is inverse of tan(x). So the domain of tan(x) is the range of arctan(x). The domain of tan(x) is {x : |x| < pi/2}

if -inf < tan(2x+1) < inf, then -pi/2 < 2x+1 < pi/2, the domain of tan(2x+1).
So -pi/2 - 1 < 2x < pi/2 -1
and -(pi + 2)/2 < 2x < (pi - 2)/2
and -(pi +2)/(2*2) < 2x/2 < (pi -2)/(2*2)
and -(pi + 2)/4 < x < (pi - 2)/4
=> |x| < (pi - 2)/4

Ans. Range of arctan(2x+1) is domain of tan(2x+1) = { x : |x| < (pi - 2)/4 }