Please help me calculate the implied domain and range

[tonyy]

Hey guys I am having trouble solving this question. Thanks in advance!

Given that cos(x) is restricted to [0,pi]. Define the implied domain and range of y=cos(tan^-1(x))

[Maths Forever]

Hey guys I am having trouble solving this question. Thanks in advance!

Given that cos(x) is restricted to [0,pi]. Define the implied domain and range of y=cos(tan^-1(x))

Let y = arctan(x) = tan^-1 (x)

Since cos(y) has a restricted domain of [0, pi], arctan(x) has range of [0, pi].

The graph of arctan(x) has horizontal asymptotes of x = pi/2 and x = -pi/2. Since the graph passes through the origin (0,0) and approaches pi/2 as x approaches infinity, implied domain of arctan(x) = [0, infinity).

Therefore, cos(arctan(x)) has implied domain of [0, infinity).

If x = 0, cos(arctan(0)) = cos(0) = 1

If x = 1, cos(arctan(1)) = cos(pi/4) = sqrt (2) /2

If x = sqrt (3), cos(arctan (sqrt (3) ) = cos (pi/3) = 1/2

Therefore, as x approaches infinity, cos(arctan(x)) approaches 0.

Therefore, the implied range of cos(arctan(x)) = (0,1].

I hope this helps!