Natural Domain of trig functions

[Loppyfoot]

Homework Statement

Find all the natural domain of the function algebraically, and confirm that your result is consistent with the graph produced by your graphing utility.
h(x) = 3/2-cosx

Homework Equations

(a) h(x) = 3/2-cosx
(b)x²-1/(x+1)

The Attempt at a Solution

Do I need to know trigonometric identities? This is just an introductory chapter, so I am guessing I do not need to know the identities. 2-cosx = 0 to find the discontinuity. But how do I find the natural domain?

 

[Cyosis]

The natural domain is the largest domain. For example you could have just 0 as domain, while valid this would not be the largest domain therefore not the natural domain. The best way to do this is to start with the domain R and then remove all points from R for which the function is not defined. Solving the equation 2-cos(x)=0 is the right approach, because that x will be the x value that will not be in the domain (keep in mind that there is more than one x value). The only 'identity' you need to know is the inverse of the cosine. If cos x=y then x=arccos y.

 

[Mark44]

Homework Statement

Find all the natural domain of the function algebraically, and confirm that your result is consistent with the graph produced by your graphing utility.
h(x) = 3/2-cosx

Homework Equations

(a) h(x) = 3/2-cosx
(b)x²-1/(x+1)

The Attempt at a Solution

Do I need to know trigonometric identities? This is just an introductory chapter, so I am guessing I do not need to know the identities. 2-cosx = 0 to find the discontinuity. But how do I find the natural domain?

It's not apparent from what you wrote, but I believe your first function is this: h(x) = 3/(2 - cos(x)). Because of the lack of parentheses, what you wrote would normally be interpreted as h(x) = 1.5 - cosx.

For the second function, do you mean x² - 1/(x + 1) or (x² - 1)/(x + 1)? I suspect that you meant the latter, but most would interpret what you wrote as the former.

 

[VietDao29]

Homework Statement

Find all the natural domain of the function algebraically, and confirm that your result is consistent with the graph produced by your graphing utility.
h(x) = 3/2-cosx

Homework Equations

(a) h(x) = 3/2-cosx
(b)x²-1/(x+1)

The Attempt at a Solution

Do I need to know trigonometric identities? This is just an introductory chapter, so I am guessing I do not need to know the identities. 2-cosx = 0 to find the discontinuity. But how do I find the natural domain?

(b) looks like a piece of cake. Do you have any troubles dealing with (b)?

For (a), yes, you need to solve for x in the equation 2 - cos(x) = 0. Well, you don't need any identity here. Big hint of the day, try to answer the following questions:

1. What's the range of cos(x)?
2. Spoiler

   From there, what's the range of 2 - cos(x)?
   (Only look here, when you are completely stuck, and find the 3rd question unanswerable)
3. Can you solve the equation 2 - cos(x) = 0 for x?