Problem: Determine the domain of the following function:
Solution: There is a general power (the root), so we have to start by rewriting it using the "e to ln" trick:
Now we are ready to look for trouble:
1. The outside logarithm requires that its argument be positive:
2. The middle logarithm requires that its argument be positive:
3. The inside logarithm requires that its argument be positive:
4. Passing to the exponential, its argument can be arbitrary, so we
look at the argument by itself. First we see the little fraction, its
denominator cannot be zero:
5. Then we have the logarithm, its argument should be positive:
6. Cosine eats anything, so we move on. The next term is
7. The logarithm requires
that is,
Now we look closer at the conditions that require further work.
Condition 1: If we raise both sides of the inequality as arguments of
Now we raise both sides as the argument of
Condition 2: If we raise both sides of the inequality as arguments of
Note that the conditions 1, 2, and 3 are of similar type and gradually less
and less restrictive, so later during the intersection phase we can disregard
the 2nd and 3rd and only consider the first,
Condition 5: This is best done by first considering
or some
shift of this interval by
Thus x must belong to the interval
Condition 6: By taking the exponential of both sides and cancelling
Now we make an intersection of all conditions
and get the answer: