Function Domain and Range finding by graphing or not

Question

When you are trying to figure out the domain and range of a function like $$\arcsin(x-3)/4$$ or $$((x-4)/(x^2-5))\ln x$$ is there an algebraic way to do it. Or is the best way to graph the equation?

I can figure out the domain/range of $$\arctan (\cos x)$$ for example from remembering what range and domain of $\arctan$ normally is. And for other functions knowing that the square root has to be of a positive number and $0$ can't be in the denominator helps but I'm a little lost with the above $2$ functions, without resorting to graphing them.

Answer 1

Re $\dfrac{\arcsin(x-3)}{4}$: we can make $x-3$ take any value in $\mathbb{R}$: what values of $x-3$ fit the domain of $\arcsin$? That'll give the domain of this function. For the range, we can put any valid value into the $\arcsin$ function by picking an appropriate $x$, so the $\arcsin$ part takes every value in the range of $\arcsin$. What does dividing by $4$ do to that range?

Re $\dfrac{x-4}{x^2-5}\ln x$: The domain is easy using the kind of rules you've described: the '$\ln x$' part restricts us to $x>0$ and division by $x^2=5$ forbids $x=\pm\sqrt{5}$; everything else is OK. The range is trickier: I could give you my careful reasoning, but I only arrived at my careful reasoning after looking at a plot :-). Generally you find all the intervals between where the function is $0$ or undefined (i.e. function potentially changes sign) and think about the range of the function in each interval.