The formula
denominator ≠ 0 · even root ≥ 0 · logarithm > 0
Why it works
The domain does not have to be guessed: it is read off the expression itself, because only three things can spoil a real calculation. A denominator cannot be zero, an even root does not take negative arguments and a logarithm needs a strictly positive one. Any other combination of sums, products and powers works on the whole line. There is no equivalent rule for the range: it depends on the whole function and generally needs the extrema studied, so here it is explored numerically and said to be an exploration.
How to solve it by hand
- Find the denominators and note where they vanish: those points leave the domain
- Find the even roots: their argument has to be greater than or equal to zero
- Find the logarithms: their argument has to be strictly positive
- For the range, walk the domain and note the largest and smallest values
What is worth knowing
Mind the difference between even and odd roots: the cube root of a negative number exists perfectly well, and only even indices restrict anything. And with a logarithm the restriction is strict: zero is out too, not only the negatives. When reading the result, look at the stretches: if the function comes back split into two or more pieces, there is a vertical asymptote in between, and the domain is a union of intervals rather than one. The exploration interval can be changed, and with fast-growing functions it is worth narrowing it to see anything.