What is the domain and range of f(x)= ln(x) + ln(1-x)?
Domain: we need to make sure that both logarithms exist, and every logarithm has the same request: its argument must be strictly positive. So, the first term,
Again, both conditions need to be satisfied, so we must consider only the numbers between
Range: As for the range, it's useful to see what happens at the borders of the domain. We have
Similarly, we have
So, our function looks like an upside-down "U". The range is thus unbounded towards negative infinity. As for the maximum, we simply need to derive:
and we have