Creating an inverse function

Question

I have this function $f(x) = x^2 + 5x + 6$

And I want to convert it to an inverse function, but I don't really know where to begin.

What I have done:

$y = x^2 + 5x + 6$

$y - 6 = x(x+5)$

But now I feel lost.

Answer 1

First you should find where the function is monotone, so as it's inverse exists .. For $f(x)$ being monotone, then then inverse is defined in $]-\infty, -2.5[$ and $]-2.5, +\infty[$. Now, we express $x$ in terms of $f(x) = x^2 + 5x + 6$, $$x^2 + 5x + (6-f(x)) = 0$$ The discriminant is $$\Delta = 1 + 4f(x)$$ The roots are $$x_{\pm} = \frac{-5 \pm \sqrt{1 + 4f(x)}}{2}$$Note that $1 + 4f(x) \geq 0$ since the minimum value of $f(x_m)$ is $-\frac{1}{4}$ attained at $x_m = -2.5$. You could easily verify that in the interval $]-\infty, -2.5[$, then inverse function is $$x_{-} = \frac{-5 - \sqrt{1 + 4f(x)}}{2}$$Also in the interval $]-2.5, +\infty[$, then inverse function is $$x_{+} = \frac{-5 + \sqrt{1 + 4f(x)}}{2}$$