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

Aug 5, 2015

Domain = $\mathbb{R}$
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 $\mathbb{R}$ 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.
$X v = - \frac{b}{2 a}$
Since this function doesn't have a $b$ term, the vertex is exactly on the y axis ($X v = 0$). Therefore, we can substitute Xv in the law of the function and find the y value that corresponds to it:
$f \left(0\right) = {0}^{2} - 4 = - 4$

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]}