Question

Let $R_{+}$ be the set of all non-negative real numbers. Show that the function $f:R_{+}\to [4,\infty )$ given by $f\left(x\right)=x^2+4$ is invertible and write the inverse of $f$.

Answer 1

Let $y=f\left(x\right)$ then $y=x^2+4$
Since $x^2\ge 0$, we get, $x^2+4\ge 4$
$\therefore$ Thus $y\ge 4\Rightarrow yϵ[4,\infty )$
$\therefore y=x^2+4\Rightarrow x^2=y-4$
$\Rightarrow x=\sqrt[\pm ]{\pm y-4}$ but $xϵR_{+}$
$\therefore x=+\sqrt{y-4}$
Define $g:[4,\infty )\to R_4$
$\Rightarrow g\left(y\right)=\sqrt{y-4}$
$\therefore gof=I_g$ and $fog=I_{[4,\infty )}$
$\therefore f$ is invertible with the the inverse given by
$f^{-1}=g\Rightarrow f^{-1}\left(y\right)=\sqrt{y-4}$.