# How do you find the domain and range of y = x^2 - x + 5?

Nov 17, 2017

The function is a polunomial, so its domain is the whole set of real numbers. $D = \mathbb{R}$.

To find the range we have to look at the formula.

The graph of the function is a parabola. The coefficient of ${x}^{2}$ is positive, so the parabola goes to $+ \infty$ as the argument goes to
$\pm \infty$, so the range is R= < q;+oo), where $q$ is the $y$ coordinate of the vertex.

To calculateit we can first calculate $x$ coordinate of the vertex (usually called $p$)

$p = \frac{- b}{2 a} = \frac{1}{2}$

Now we can calculate $q$ by substituting $p$ to the function's formula:

$q = f \left(\frac{1}{2}\right) = {\left(\frac{1}{2}\right)}^{2} - \left(\frac{1}{2}\right) + 5 = 4 \frac{3}{4}$

Now we can write the range: