What is the domain and range of: y=5x^2+2x-1?

www.wolframalpha.com/input/?i=y%3D5x^2%2B2x-1

That is a parabola opening up. All the properties of a parabola are listed at
en.wikipedia.org/wiki/Parabola

A function is any process that gives exactly one ooutput for each qualilfied input. Domain means qulified inputs and range means possible outputs. For your example the domain is all numbers and the range is y greater than or equal to -1.