I have to prove that $f \colon x \mapsto e^{4x} + x^5 + 2$ ($f\colon \mathbb{R}\to\mathbb{R}$) is bijective. The argument given in the solution is that since the first two summands of the image is a bijective function of $x$, then so is $f$. Nonetheless, this "proof" doesn't seem at all rigorous to me, since there are many counterexamples to this argument.
So I proved that $f$ is strictly increasing by looking at its derivative, thus injective, and that every $c \in \mathbb{R}$ has at least one preimage, applying Bolzano's theorem to $f(x) - c$ and evaluating its limit at $-\infty$ and $+\infty$, and so, demonstrating $f$ to be surjective. In consequence, $f$ is bijective.
I am much more pleased about this proof than the one given in the solutions, but I want to know if I missed something, or if my hypothesis are insufficient. I gave an example to illustrate my argument, but the question I want to ask in the general form is in the title. Also, I would like to know whether the converse holds as well.
Thanks.
Update: I've been thinking about this, and realized that only monotonicity and continuity (together with unboundedness, as Mariano pointed out) are necessary for bijectiveness, and derivability only helps to prove monotonicity. Is this correct?