# What is the domain and range of f(x) = x^2 + 4x – 6?

Oct 1, 2015

Domain: $\mathbb{R}$
Range: $\mathbb{R} \ge - 10$

#### Explanation:

$f \left(x\right) = {x}^{2} + 4 x - 6$
is valid for all Real values of $x$
and therefore the Domain is all Real values i.e. $\mathbb{R}$

To determine the Range, we need to find what values of $f \left(x\right)$ can be generated by this function.
Probably the simplest way to do this is to generate the inverse relation. For this I will use $y$ in place of $f \left(x\right)$ (just because I find it easier to work with).

$y = {x}^{2} + 4 x - 6$

Reversing the sides and completing the square:
$\textcolor{w h i t e}{\text{XXX}} \left({x}^{2} + 4 x + 4\right) - 10 = y$

Re-writing as a square and adding $10$ to both sides:
$\textcolor{w h i t e}{\text{XXX}} {\left(x + 2\right)}^{2} = y + 10$

Taking the square root of both sides
$\textcolor{w h i t e}{\text{XXX}} x + 2 = \pm \sqrt{y + 10}$

Subtracting $2$ from both sides
$\textcolor{w h i t e}{\text{XXX}} x = \pm \sqrt{y + 10} - 2$

Assuming that we are restricted to Real values (i.e. non-Complex), this expression is valid provided:
$\textcolor{w h i t e}{\text{XXX}} y \ge - 10$
$\textcolor{w h i t e}{\text{XXXXXX}}$(otherwise we would be dealing with the square root of a negative value)