Consider `f:R_+->[-9,oo[` given by `f(x)=5x^2+6x-9`. Prove that `f` is invertible with `f^(-1)(y)=(sqrt(54+5y)-3)/5`

Explanation

`f(x)=5 x^{2}+6 x-9`


Let `y=5 x^{2}+6 x-9`


`=5(x^{2}+frac{6}{5} x-frac{9}{5})`


`=5(x^{2}+2 times x times frac{3}{5}+frac{9}{25}-frac{9}{25}-frac{9}{5})`


`=((x+frac{3}{5})^{2}-frac{9}{25}-frac{9}{25})`


`=(x+frac{3}{5})^{2}-frac{9}{5}-9`


`Rightarrow 5(x+frac{3}{5})^{2}-frac{54}{5}`


`Rightarrow frac{5 y+54}{25}(x+frac{3}{5})^{2}`


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