Show that $f$ has an inverse The function $f$ is given on the interval $[0,\infty)$ by
$$f(x)=\sqrt{x^5+4x^3+3x+1}$$
Show that $f$ has an inverse.

$f'(x)=\frac{5x^4+12x^2+3}{2\sqrt{x^5+4x^3+3x+1}}>0$ for all $x$, so $f$ is injective. How do I show that $f$ is surjective?

You've shown that $f$ is strictly increasing, which indeed lets you conclude that $f$ is injective on $[0,\infty)$. But this is sufficient to guarantee the existence of an inverse $f^{-1}:f[0,­\infty) \to [0,\infty)$.
Indeed—and this has nothing to do with analysis, just plain old set theory—any function $f:A \to B$ between two sets is surjective onto its image. In other words, $f:A \to f(A)$ is surjective, by definition of surjectivity. If in addition $f$ is injective, then $f:A \to f(A)$ is bijective, and so it has an inverse $f^{-1}:f(A) \to A$.
Now whether you want to compute the range $f[0,\infty) = [1,\infty)$ explicitly is up to you.
