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

Jul 1, 2015

Domain: all real $x$;
Range: $y \ge - 3$

#### 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:
${x}_{v} = - \frac{b}{2 a} = 0$
${y}_{v} = - \frac{\Delta}{4 a} = - \frac{{b}^{2} - 4 a c}{4 a} = - 3$
Where your equation ${x}^{2} - 3$ is in the form $a {x}^{2} + b x + c$ with:
$a = 1$
$b = 0$
$c = - 3$
So the range (the possible $y$ values of your function) will be all the values $y \ge - 3$

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