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

Nov 30, 2017

Domain: $x \in \mathbb{R}$
Range: $f \left(x\right) \in \left[- 4 , + \infty\right)$

#### Explanation:

$f \left(x\right) = {x}^{2} - 2 x - 3$ is defined for all Real values of $x$
therefore the Domain of $f \left(x\right)$ covers all Real values (i.e. $x \in \mathbb{R}$)

${x}^{2} - 2 x - 3$ can be written in vertex form as ${\left(x - \textcolor{red}{1}\right)}^{2} + \textcolor{b l u e}{\left(- 4\right)}$ with vertex at $\left(\textcolor{red}{1} , \textcolor{b l u e}{- 4}\right)$
Since the (implied) coefficient of ${x}^{2}$ (namely $1$) is positive, the vertex is a minimum
and $\textcolor{b l u e}{\left(- 4\right)}$ is a minimum value for $f \left(x\right)$;
$f \left(x\right)$ increases without bound (i.e. approaches $\textcolor{m a \ge n t a}{+ \infty}$) as $x \rightarrow \pm \infty$
so $f \left(x\right)$ has a Range of $\left[\textcolor{b l u e}{- 4} , \textcolor{m a \ge n t a}{+ \infty}\right)$