confused about domain and range

How do you find these out algebraically given a function?

It'd be appreciated if you could do an example with an exponential (2^x) function, a root function, a reciprocal function, a cubic function, a semi-circle function, a quadratic function, absolute value function, conics (ellipse, hyperbola, parabola of the form y=1/4px^2, etc.) and whatever else.

Or, if you could just explain how it's done in general. My textbook has no information on this and my teacher didn't explain any of this. I googled around, but they all use different methods. Could I please get *one* method that works for all cases?

Here's a few examples that I just came up with that I have no idea how to solve.

y = 9x^2 / (25-x^2)

y = sqrt(4x^2-9)

y = (4x^2-9)/x^2

y = 5 / sqrt(3-x)

y = 4x^2+6x+6

y = |x-3|+6