# What is the domain and range for g(x)= x^2 - 3x?

Jun 5, 2015

$g \left(x\right)$ is well defined for all $x \in \mathbb{R}$ so its domain is $\mathbb{R}$ or $\left(- \infty , \infty\right)$ in interval notation.

$g \left(x\right) = x \left(x - 3\right) = \left(x - 0\right) \left(x - 3\right)$ is zero when $x = 0$ and $x = 3$.

The vertex of this parabola will be at the average of these two $x$ coordinates, $x = \frac{3}{2}$...

$g \left(\frac{3}{2}\right) = {\left(\frac{3}{2}\right)}^{2} - 3 \left(\frac{3}{2}\right) = \frac{9}{4} - \frac{9}{2} = - \frac{9}{4}$

As $x \to \pm \infty$ we have $g \left(x\right) \to \infty$.

So the range of $g \left(x\right)$ is $\left[- \frac{9}{4} , \infty\right)$

graph{x^2-3x [-10, 10, -5, 5]}