How do you find the range and domain of a function algebraically?

You find the domain of the function by finding all numbers that can't go into the equation. Usually all that prevents this is whether or not it contains a fraction with x on the bottom or a x under a square root symbol. 

The range is found by finding the vertex and noting that it is the minimum or maximum (depending on direction) of the graph. You can find the vertex by using -b/2a for the x value and then plugging in to get y value.