Question

How do you find the domain and range for `F(x) = x^2 - 3`?

Answer 1

Domain: all real `x`;
Range: `y>=-3`

Explanation:

Your function is a Quadratic, and can be represented graphically by a Parabola.
Your function can accept any value of `x` so that the domain will be all the Real `x`.

The range is a little bit tricky...!

Your function has a minimum value (the vertex of your parabola) whose `y` value gives an idea of the range (it is the lowest point attained by your function).

The coordinates of the vertex can be found as:
`x_v=-b/(2a)=0`
`y_v=-Delta/(4a)=-(b^2-4ac)/(4a)=-3`
Where your equation `x^2-3` is in the form `ax^2+bx+c` with:
`a=1`
`b=0`
`c=-3`
So the range (the possible `y` values of your function) will be all the values `y>=-3`

Graphically:
graph{x^2-3 [-8.89, 8.89, -4.444, 4.445]}