Let `f: NvecS` be a function defined as `f(x)=4x^2+12 x+15.` Show that `f: NvecS ,` where `S` is the range of `f,` is invertible. Also find the inverse of `f`

Explanation

`f(x)=4 x^{2}+12 x+15=y(u t)`


`D=144-4 times 4 times 15<0`


`therefore f(x)` will always be `+` ve


Hence it u invertible.


`4 x^{2}+12 x+15-y=0`


`x=frac{-12 pm sqrt{144-4 times 4 times(15-y)}}{8}`


`x=frac{-12 pm 4 sqrt{9-15+y}}{8}`


`x=frac{-3 pm 2 sqrt{y-6}}{2} rightarrow` inverse