State the doman and range of the relation y= Arc tan x

Question

Domain: All real numbers
Range= -pi/2 < y < pi/2

Answer 1

You can get into great arguements with High School math teachers on the range.

I would put
-inf<y<inf

But I have to tell you, many will argue that your range is "correct".

They don't want to recognize that y could be -400PI

Answer 2

Haha, okay, but since it's a multiple choice question and that's the only choice...I think I'll go with that for now.

Answer 3

bobpursley would have no argument from this "old-school" high school math teacher.

I agree with him, with one exception

y= Arc tan x is the same as x = tan y
or
y = Arc tan x is the inverse of y = tanx

the domain of y = tanx is the set of real numbers, except x = k(pi/2), where k is an integer.
(there are asymptotes at those values)

so the inverse, or y = arctan x, would have the same range

since the inverse of a relation results in a reflection in the line y=x, the original vertical asymptotes turn into horizontal anymptotes and the tangent curves are layered between those horizontal asymptotes.
I believe the multiple choice is given the way it is, because calculators would only give answers in -pi/2 < y < pi/2

Answer 4

There is more than one choice for an inverse of the tan function. But you need to make a definite choice. The standard arctan function has range from -pi/2 to pi/2.

This is important if you consider these inverse function on the complex plane. If you define the logarithmic function by making a choice for the branch cut across which the argument of log(z) jumps by minus 2 pi i, then all the inverse functons are fixed.