# How do you find the domain and range of f(x)=(x-1)^2 +2?

Oct 17, 2016

The domain is all real numbers $\mathbb{R}$ or in interval notation $\left(- \infty , \infty\right)$.

The range is $y \ge 2$ or in interval notation $\left[2 , \infty\right)$.

#### Explanation:

$f \left(x\right) = {\left(x - 1\right)}^{2} + 2 = \textcolor{b l u e}{1} {\left(x - \textcolor{red}{1}\right)}^{2} + \textcolor{red}{2}$

The easiest way to find the domain and range of a quadratic function is to look at the graph.

The general equation of a parabola in vertex form is
$y = a {\left(x - h\right)}^{2} + k$ where $\left(h , k\right)$ is the vertex.

A positive $a$ means the parabola is upward facing (U shaped) and a negative $a$ means it is downward facing (an upside down U shape).

The vertex of this example is then $\left(\textcolor{red}{1} , \textcolor{red}{2}\right)$.
$a = + \textcolor{b l u e}{1}$ so the shape is an upward facing parabola.

The domain is found by considering all the possible values of $x$.
Looking at the graph, you can see that $x$ goes all the way from negative infinity to positive infinity. The domain can be expressed as all real numbers $\mathbb{R}$, or in interval notation, $\left(- \infty , \infty\right)$.

The range is found by considering all the possible values of $y$.
There are no values of $y$ below $y = 2$, so $y \ge 2$. In interval notation, the range is $\left[2 , \infty\right)$.

graph{(x-1)^2+2 [-10.12, 9.88, -2.4, 7.6]}