# [SOLVED] Solving for inverse functions: f(x)=x^2-5x-6

#### Busykid

I'm trying to find the inverse function of f(x)=x^2-5x-6

All help is appreciated!

Edit: Figured it out!

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#### HallsofIvy

MHF Helper
In case others are interested- the first thing I would do is "complete the square". $$\displaystyle y= x^2- 5x- 6= x^3- 5x+ \frac{25}{4}- \frac{25}{4}- 6$$$$\displaystyle = \left(x- \frac{5}{2}\right)^2- \frac{49}{4}$$. That tells me that the vertex of this parabolic graph is at $$\displaystyle \left(\frac{5}{2}, \frac{49}{4}\right)$$. So, while the entire function does NOT have a true "inverse" I can restrict the domain to get two functions that have inverses. From $$\displaystyle y= \left(x- \frac{5}{2}\right)^2- \frac{49}{4}$$, $$\displaystyle \left(x- \frac{5}{2}\right)^2= y+ \frac{49}{4}$$ and then $$\displaystyle x= \frac{5}{4}\pm\sqrt{y+ \frac{49}{4}}$$. The part with $$\displaystyle x< \frac{5}{4}$$ is given with the "-" and that with $$\displaystyle x> \frac{5}{4}$$ is given with the "+".

Finally, swapping "x" and "y", the two inverses are given by $$\displaystyle y= \frac{5}{4}+ \sqrt{x+ \frac{49}{4}$$ and $$\displaystyle y= \frac{5}{4}- \sqrt{x+ \frac{49}{4}$$.

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