Expert: Paul Klarreich - 7/28/2009

Question
When approaching a domain/range question, it is difficult to determine the domain/range when there are more than one type of function in the equation?
For example:Find the domain/range for the function f(x)=ln(cosx)?
There is a natural log. and a trig function concocted into one.
Simply, how do you solve domain/range questions when it is dealing with different functions other than simply find the dom/ran of cos x?

Answer
Questioner: Kilonu
Country: United States
Category: Advanced Math
Private: No
Subject: Function/Domain/Range
Question: When approaching a domain/range question, it is difficult to determine the domain/range when there are more than one type of function in the equation?
For example:Find the domain/range for the function f(x)=ln(cosx)?
There is a natural log. and a trig function concocted into one.
Simply, how do you solve domain/range questions when it is dealing with different functions other than simply find the dom/ran of cos x?
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Hi, Kilonu,

I think you mean 'composed'.

Anyway, nobody promised this would be easy.  

When you have a composition of two functions:

f(x) =  h( g(x)), you must:

1. Determine and exclude any 'bad' numbers for g(x).  If g(x) is undefined at  x = x0, then  x0 is out.

2. Find any numbers in the RANGE of g(x) that are 'bad' numbers for  h(x).

In this case,  f(x) = ln(cos x),

the 'inner' function, cos x, is defined for all x, so no problem there, but  the 'outer' function, ln x, is defined only for positive values.

So any x for which  cos x <= 0 is 'bad', and:
  any x for which  cos x >  0 is 'good':


Now cos x > 0 for  0 <= x < pi/2 and  for 3pi/2 < x < 2pi

So the domain is  0 <= x < pi/2 and 3pi/2 < x < 2pi

Oh, yes:  +- 2n pi for all of those.