Answer
See proof
Work Step by Step
From the attached image, it is clear that for every $y$-value, there exists an unique $x$-value associated with the function $f(x)=x^{6}+3x+5, x\ge0$
$\implies$ The function is inversible.
According to the Theorem which states that if $f$ is a one-to-one function that is continuous at each point of its domain, then $f^{-1}$ is continuous at each point of its domain, we conclude that as $f$ is continuous on its range (because it is a polynomial),
$\implies$ The inverse is continuous on $[5,\infty)$.
