# What is the domain and range of the function y = x^2- x + 5?

Mar 10, 2018

Domain: $\left(- \infty , \infty\right)$ or all reals
Range: $\left[\frac{19}{4} , \infty\right)$ or $\text{ } y \ge \frac{19}{4}$

#### Explanation:

Given: $y = {x}^{2} - x + 5$

The domain of an equation is usually $\left(- \infty , \infty\right)$ or all reals unless there is a radical (square root) or a denominator (causes asymptotes or holes).

Since this equation is a quadratic (parabola), you would need to find the vertex. The vertex's $y$-value will be the minimum range or the maximum range if the equation is an inverted parabola (when the leading coefficient is negative).

If the equation is in the form: $A {x}^{2} + B x + C = 0$ you can find the vertex:

vertex: $\left(- \frac{B}{2 A} , f \left(- \frac{B}{2 A}\right)\right)$

For the given equation: $A = 1 , B = - 1 , C = 5$

$- \frac{B}{2 A} = \frac{1}{2}$

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

$f \left(\frac{1}{2}\right) = \frac{1}{4} - \frac{2}{4} + \frac{20}{4}$

$f \left(\frac{1}{2}\right) = \frac{19}{4} = 4.75$

Domain: $\left(- \infty , \infty\right)$ or all reals
Range: $\left[\frac{19}{4} , \infty\right)$ or $\text{ } y \ge \frac{19}{4}$

graph{x^2-x+5 [-25.66, 25.66, -12.82, 12.83]}