Question

How do you find the domain and range of `y = x^2 - 4`?

Answer 1

Domain = `RR`
Range = `[-4, +oo[`

Explanation:

This quadratic function does not have a domain restriction, therefore all values of x are possible. Domain can be written as `RR` or `]-oo, +oo[`.

In order to find the range, you must find the y coordinate of the vertex. There are many ways to do that, but I prefer to start by finding the x value of the vertex before finding y.
`Xv=-b/(2a)`
Since this function doesn't have a `b` term, the vertex is exactly on the y axis (`Xv=0`). Therefore, we can substitute Xv in the law of the function and find the y value that corresponds to it:
`f(0)=0^2-4=-4`

Every quadratic function has the shape of a parabola. Since `a` is positive, the parabola has its concave up. Therefore, it's not possible to have a y value below the vertex of the function.
The range of the function is `[-4, +oo[`

Check out the graph of the function. If you have a doubt, you can always try sketching the graph.
graph{x^2-4 [-10, 10, -5, 5]}