# How do you find the domain and the range of the function f(x)= x^2 - 2x -3?

Mar 26, 2018

Domain: $\left(- \infty , \infty\right)$ Range: $\left[- 4 , \infty\right)$

#### Explanation:

In general, the domain, or $x$ values which yield an output $f \left(x\right) ,$ of a polynomial function is all real numbers, denoted by $\left(- \infty , \infty\right)$ in interval notation.

Range becomes more specific. This is a quadratic function. In general, the range, or $y -$values for which the function exists, of a quadratic with vertex at $\left(h , k\right)$ is $\left[k , \infty\right)$, OR $\left(- \infty , k\right]$ if the parabola is inverted (it isn't in this case -- we don't begin with $- {x}^{2}$).

So, let's find the vertex.

We have $f \left(x\right) = a {x}^{2} + b x + c = {x}^{2} - 2 x - 3 , a = 1 , b = - 2 , c = - 3$

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

$k = f \left(h\right) = f \left(1\right) = 1 - 2 - 3 = - 4$

The range is then $\left[- 4 , \infty\right]$