Whenever you learn a basic function, you should understand its domain. For example, the domain of f(x) = 1/x is all real numbers except 0, because you cannot divide by zero (do you know why?). If you're working with the real numbers, and not the complex numbers, then sqrt(x) has domain non-negative numbers, because you can't take the square root of a negative number (do you know why?).

You should understand the domain of each basic function that you learn about.

Then when you combine functions, you need to reason about the combination to figure out the domain of the overall thing. For example, consider f(x) = 1/(sqrt(x - 3) - 4). There are two possible issues here. We're combining several functions, but in particular, 1/x and sqrt(x), so we need to look at that. We know that sqrt(x) has domain non-negative real numbers. But here we have sqrt(x - 3). So if we want to only sqrt non-negative numbers, we need x - 3 to be non-negative. That means x must be greater than or equal to 3.

So if x >= 3, then we will have a valid input into sqrt. Next, let's look at 1/x. Here x must not be 0. But we're actually doing 1/(sqrt(x - 3) - 4), so we're dividing by sqrt(x - 3) - 4, which means sqrt(x - 3) - 4 must not be 0. sqrt(x - 3) - 4 is 0 when x = 19.

So x = 19 will cause us problems with 1/x, and all xs less than 3 will cause us problems with sqrt(x). So the overall domain is all real numbers greater than or equal to 3, except 19.