# How do you find the inverse of f(x) = x^2 +x?

Nov 12, 2015

${f}^{-} 1 \left(x\right) = - \frac{1}{2} \pm \sqrt{x + \frac{1}{4}}$

#### Explanation:

To find an inverse, switch x and y (or f(x)).

For simplicity I will rewrite f(x) as y...
$y = {x}^{2} + x$
switch...
$x = {y}^{2} + y$
now solve for y
It looks like we need to complete the square...
$x + \frac{1}{4} = {y}^{2} + y + \frac{1}{4}$
$x + \frac{1}{4} = {\left(y + \frac{1}{2}\right)}^{2}$
$\pm \sqrt{x + \frac{1}{4}} = y + \frac{1}{2}$ Don't forget the $\pm$!
$- \frac{1}{2} \pm \sqrt{x + \frac{1}{4}} = y$ Don't forget the $\pm$!