Exercise
Determine the domain of the function:
Final Answer
Solution
Let’s find the domain of the function:
Because there is a denominator, the denominator must be different from zero:
Also, there is a ln function, so we need the expressions inside the ln to be positive:
Let’s look at inequality. Sin function is positive in the first half of its cycle. Therefore, in the first cycle we receive:
But it has endless cycles. We express this with parameter k (integer). Adding the cycle of sin function in both sides results in
Now one see that to isolate x one have to break up into cases:
where k = 0 we get
From this inequality we get that for x> 0 (positive) we get
But for x <0 (negative) we get
Meaning, there is no solution.
Remember that x is different from zero, otherwise we get a denominator function equal to zero, which is not defined.
Let’s move to find thedomain for k that is different from zero. Recall the inequalities we received:
Again, to divide by x or k one have to determine whether they are positive or negative. Therefore, when k> 0 (positive) then for x <0 (negative) we get
That is, there is no solution. But for x> 0 (positive) we get
And when k <0 then for x> 0 we get
That is, there is no solution. But for x <0 we get
And the final answer is the union of the 3 results we got: for k = 0, k> 0 and k <0. Therefore, we get
and
and
Have a question? Found a mistake? – Write a comment below!
Was it helpful? You can buy me a cup of coffee here, which will make me very happy and will help me upload more solutions!