Inverse Functions
When writing an inverse function, you write the function in the form $y = …$ and interchange the xs and ys. And change the form of the equation such that y is alone in one side of the equation. This does not always apply for polynomials. In the case of second-order polynomials for example, you use the formulae $\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ . An example of taking the inverse of the function $f(x) = x^2 - 4x + 7$ for $x \ge 2$.
$$y = x^2 - 4x + 7$$ $$x = y^2 - 4y + 7$$ $$0 = y^2 - 4y + 7 - x$$ $$a=1, b=-4, c=7-x$$ $$y = \frac{4 \pm \sqrt{16 - 4(7-x)}}{2}$$ $$y = \frac{4 \pm 2\sqrt{4 - 7+x}}{2}$$ $$y = 2 \pm \sqrt{x-3}$$ Two possible options: $$y = 2 + \sqrt{x-3}$$ $$y = 2 - \sqrt{x-3}$$ Since when we take the inverse functions, we switch the domain and the range, we pick the function with the range $y \ge 2$, so the first one. Therefore:
$$f^{-1}(x) = 2 + \sqrt{x-3}$$