State the doman and range of the relation y= Arc tan x
Domain: All real numbers
Range= -pi/2 < y < pi/2
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
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.
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
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.
To determine the domain and range of the relation y = Arc tan x, we need to consider the properties and limitations of the arctangent function.
The domain represents all the possible input values for which the relation is defined. For the arctangent function, it is defined for all real numbers. This means that any real number can be substituted for x in the relation y = Arc tan x.
Therefore, the domain of y = Arc tan x is all real numbers.
The range represents all the possible output values of the relation. For the arctangent function, it has a limited range between -π/2 and π/2. This means the output values of y are restricted to values between -π/2 and π/2.
Therefore, the range of y = Arc tan x is -π/2 < y < π/2.