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

Feb 3, 2016

Domain = $\mathbb{R}$.
Range = $\left[4.75 , \infty\right)$

#### Explanation:

This is a 2nd degree quadratic equation so its graph is a parabola with arms going up since the coefficient of ${x}^{2}$ is positive, and turning point (minimum value) occurring when $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$, that is when $2 x - 1 = 0$, whence $x = \frac{1}{2}$.
But $y \left(\frac{1}{2}\right) = 4.75$.

Hence the domain is all allowed input x-values and is thus all real numbers $\mathbb{R}$.
The range is all allowed output y values and is hence all y-values bigger than or equal to $4.75$.

The plotted graph verifies this fact.

graph{x^2-x+5 [-13.52, 18.51, -1.63, 14.39]}