K here is an example question (there are 4 in total but I only need to learn how to do one then I can do the rest):

ln(cosh x)

I have to find the largest possible domain for this.

Now I was told that Hyperbolics are similar to exponentials, and Inverse Hyperbolics are similar to Natural Logarithms. Here I have Ln and a normal Hypberbolic, so I don't think this helps me much.

If I look on the graph of cosh x, I can see that the minimum range is 1 < y < infinity.
The Domain is all real values of x.

Lastly, I know that cosh x is in fact and exponential function in the form: 0.5(e^x + e^-x)

I cannot simply take ln's of both sides because there is a half blocking the way :s...

Any ideas on how to tackle this problem guys?

ln (cosh x) is a function of a function. That is the range of the first function becomes the domain of the second function (where possible) so you need to :

1. Identify the domain of cosh x which is
-infinity < x < infinity

2. Identify the range of cosh x which is
1 =< y < infinity

3. Identify the domain of ln x which is
0 < x < infinity

now since the range of cosh x lies entirely within the domain of ln x, then the largest possible domain of the combined function is the domain of the first function, cosh x

i.e. domain is -infinity < x < infinity

Now if the range of the first function lies partly outside the domain of the second function you need to consider what part of the original domain matches the part of the range that lies within the domain of the second function.

As an example consider ln (sinh x).

1. the domain of sinh x is
-infinity < x < infinity

2. the range of sinh x is
- infinity < y < infinity

3. the domain of ln x is
0 < x < infinity

Here we have part of the range of sinh x outside the domain of ln x ( values =< 0) so we need to limit the domain of sinh x to that part which maps into the range 0 < y < infinity. Looking at the sinh x function you can see that sinh x is positive only when x is positive so then the largest possible domain of the combined function is limited to
0 < x < infinity.

Hope this helps!

This is bang on the point, explained very well and was extremely helpful. I just came.

Thank you.