# What is the domain and range of g(x)=2x^2-x+1?

Feb 10, 2016

Domain: $\mathbb{R}$
Range: $\mathbb{R} \ge \frac{7}{8}$

#### Explanation:

$g \left(x\right) = 2 {x}^{2} - x + 1$ is defined for all Real values of $x$
So Domain $g \left(x\right) = \mathbb{R}$

$g \left(x\right)$ is a parabola (opening upward)
and we can determine its minimum value by re-writing its expression in vertex form:

$2 {x}^{2} - x + 1$
$= 2 \left({x}^{2} - \frac{1}{2} x \textcolor{b l u e}{+ {\left(\frac{1}{4}\right)}^{2}}\right) + 1 \textcolor{b l u e}{- \frac{1}{8}}$
$= 2 {\left(x - \frac{1}{4}\right)}^{2} + \frac{7}{8}$
$\textcolor{w h i t e}{\text{XXXXXXXXX}}$with vertex at $\left(\frac{1}{4} , \frac{7}{8}\right)$

So the Range $g \left(x\right) = \mathbb{R} \ge \frac{7}{8}$
graph{2x^2-x+1 [-2.237, 3.24, -0.268, 2.47]}