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

For different values of $x$ different values of $y$ are given.

Hence one one

Co domain$=[4,\infty )$

For every $x\in R^{+},range=[4,\infty )$

$x^2+4$ has least value as $x=0,y=4$ from graph.

For all remaining values of $x$, $y>4$

range=[4,\infty ) is equal to codomain

So $f$ is one one and onto.

Inverse exists

$y=x^2+4$

$x=\pm \sqrt{y-4}$

$x\in R^{+},x=\sqrt{y-4}$

$f^{-1}\left(x\right)=\sqrt{x-4}$ is the inverse of $f$