Question

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

Answer 1

$f\left(x\right)=4x^2+12x+15=y\left(ut\right)$

$D=144-4\times 4\times 15<0$

$\therefore f\left(x\right)$ will always be +ve

Hence it u invertible.

$4x^2+12x+15-y=0$

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

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

$x=\frac{-3\pm 2\sqrt{y-6}}{2}\to$ inverse