Find the largest domain and the range for ln(cos x). How would you define the inverse function?

Question

Find the largest domain and the range for ln(cos x). How would you define the inverse function?

Answer 1

what Josh said is basically correct. He did, however overlook one thing. If you want the largest domain for the original function, you must consider all values where cos x > 0. (3pi/2,5pi/2) is also part of the domain. So is (7pi/2, 9pi/2).....all the angles coterminal with the angles given in Josh's original description. The general solution should be (-pi/2 + 2n*pi, pi/2 + 2n*pi) where n is any integer.

Answer 2

When working with a function within a function, it is important to note that the range of the inner function must not include values that are not in the domain of the outer function. The range of Cos(x) is [-1,1]. The domain of the natural logarithm is (0,+∞). Therefore, our Cos(x) must be restricted such that the range is greater than zero. So the domain must be restricted to (-π/2,π/2). Note that the interval is exclusive because ln(x) is not defined at x=0. 

 

Now, Since the function ln(x) is defined everywhere on the restricted domain interval, the function ln(cos(x)) is continuous on this interval. Because we know this and because we know that ln(x) is one-to-one and onto (meaning that for every x there is only one y and for every y there is only one x), we can determine the range of the function as being the limit of ln(x) as x approaches 0 and as x approaches 1. Evaluating these limits gives the range as being

(-∞,0).

 

Now for the final portion of this question. ln(cos(x)) does not have an inverse defined across the entire domain. We must restrict the domain such that there is one element in the range for every element in the domain and vice versa.

 

For instance, when restricting the domain of the sine function to determine the domain and range of arcsine, we use [-π/2,π/2] because, on this interval, sin(x) is one-to-one and onto.

 

The restricted domain of the inner function, cos(x), is [0,π], but we don't want to consider domain values that return a negative or zero range value. So we can further restrict the domain to [0,π/2). By consequence, the range of the function is restricted to [0,-∞). On these intervals, ln(cos(x)) is both one-to-one and onto. Now we can swap the domain with the range to determine the domain and range of the inverse. 

 

The inverse of ln(cos(x)), (arccos(e^x)), is going to have a domain of [0,-∞) and a range of [0,π/2).