I don't think one exists, but the author of my text says it should. Here's why I don't such an $f$ exists. Suppose $f$ didn't have a global inverse, clearly $f$ is not injective for else we could define $$g:\text{Im}(f)\to \mathbb{R}$$ by $$y\mapsto x$$ where $f(x)=y.$ If $f$ is not injective, then $f(a)=f(b)$ for some $a,b\in\mathbb{R}$ distinct. Then by the Mean Value Theorem there exists $c$ with $$0=\frac{f(a)-f(b)}{a-b}=f'(c).$$ By contrapositive if $f'$ is non vanishing, then $f$ is injective, hence globally invertible.
Did I perhaps make a mistake, or is the author wrong in this case?
You are correct that a differentiable function $f:\mathbb{R}\to\mathbb{R}$ whose derivative never vanishes must be injective. In fact, Darboux's theorem guarantees that $f'(x)$ cannot change sign, so it is either positive everywhere or negative everywhere, which implies that $f$ is a strictly monotone function, either strictly decreasing or strictly increasing, so it is injective indeed.
So either the author of your book means that the range of the function is not $\mathbb{R}$, or, if she/he does, it is a mistake.