How to find the inverse to $f(x)= x^2 - 6x + 11 $

Question

If the inverse exists, how do I find the inverse to this function:

$$ f(x)= x^2 - 6x + 11 $$

with $x \le 3$

Stuck at the quadtric formula. I think i have got the right answer which is $x = 3 ± \sqrt{y-2}$ ? But it doesnt seem right.

Answer 1

If the inverse exists, you just write $y=x^2-6x+11$ and use the quadratic formula to get $x$ in terms of $y$. To see if it exists, you need to ensure that for a given $y$ there is only one $x$. The obvious threat is the $\pm$ sign in the quadratic formula.

Answer 2

You don't need the quadratic formula, just complete the square! You start with $$ y = x^2 - 6x + 11 $$ or in other words $$ x^2 - 6x + 11 - y = 0 $$ Now your goal is to write that as $(x + a)^2 + \ldots = 0$ for some $a$. Observe that by expanding that square you get $x^2 + 2ax + \ldots$. Matching that to your original equation shows that you have to pick $a=-3$. That produces the correct coefficients for $x^2$ and $x$, so all you need to do is correct for the differing constant term. You get $$ (x - 3)^2 + 2 - y = 0 $$ which via simple algebra yields $$ x = 3 \pm \sqrt{y - 2} $$

Note that this always works! If the coefficient of $x^2$ in your equation isn't $1$, just divide the whole equation by the coefficient before you start. Once you've praticed this square completion a few times, you'll be at least as fast as with the formula, and you won't have to remember the formula anymore.