find the range and domain

Question

The range and domain of f(x)=sin^-1x+cos^-1x+tan^-1x is

 

 

Answer 1

I agree with the domain.

 

But, if you look at the derivative of this function you will see that there is no point at which the slope is 0 across this domain. Therefore there is no relative minimum nor maximum.

Therefore the minimum and maximum must be at the endpoints.

 

If you evaluate the function f(-1) you get a value of pi/4 and at f(1) you get a value of 3pi/4.

 

So the range is [pi/4,3pi/4]

Answer 2

the range of the inverse sine is -pi/2 to pi/2, inclusive

the range of inverse cosine is 0 to pi, inclusive

the range of inverse tangent -pi/2 to pi/2, exclusive of its endpoints, so

the range of f(x) is (0, pi/2) which is the INTERSECTION of the three ranges of its individual terms.

 

the domain of the overall function is [-1, 1] because a sine (whose angle the inverse sine 'seeks') can be anything between -1 and 1, inclusive.  The domain of tangent is any real number, but the 'weakest link' rule applies to the overall f(x).