If one adds the condition of honesty (that $|x| \leq poly(|f(x)|)$, as pointed out by Dorais in the comments), then this is *precisely* the question of whether (complexity-theoretic, as opposed to cryptographic) one-way functions exist. They exist if and only if $\mathsf{P} \neq \mathsf{UP}$ (Grollman & Selman and Ko). Whether $\mathsf{P} = \mathsf{UP}$ or not is a well-known open complexity question equivalent to the OQ.

That being said, there are many *candidates* for such functions, such as those that are used in cryptography. For example, integer multiplication is a candidate - it is known to be computable in polynomial-time, it is honest, if we restrict to multiplication of pairs $(a,b)$ where $a < b$ and $a,b$ are both prime, then it is injective, and its inverse is not known to be computable in polynomial time. Another example comes from Tensor Isomorphism: given a tensor $T \in \mathbb{F}^n \otimes \mathbb{F}^m \otimes \mathbb{F}^p$ and $g \in GL_n \times GL_m \times GL_p$, consider the function $(T,g) \mapsto (T, g \cdot T)$. This function is honest and injective, but inverting it is TI-complete. This was suggested as the basis for a bunch of cryptographic primitives in Ji-Qiao-Song-Yun.

Lastly, some results are known regarding other complexity classes, or weakening the one-to-one assumption. For example:

Say that a function $f$ (not necessarily one-to-one) is "invertible in polynomial time" if there is a poly-time-computable function $g$ such that $f(g(y)) = y$ for all $y \in \text{Image}(f)$. If every honest poly-time $f$ had such an inverse, then $\mathsf{P} = \mathsf{NP}$. (Proof: consider the function $f(\varphi,x) = \varphi$ if $\varphi$ is a 3cnf formula and $\varphi(x)=T$, and otherwise $f(\varphi,x)=0$. An inverse to this function lets you compute satisfying assignments to 3cnf's.)

Even if you only ask that $f$ have an inverse in $\mathsf{FP}^{\mathsf{NP} \cap \mathsf{coNP}}$ (aka $\mathsf{NPSV}_t$), if all honest poly-time $f$ have such an inverse, the polynomial hierarchy collapses to its second level Hemaspaandra-Naik-Ogihara-Selman. (Side note: this is related to one of my favorite open oracle questions. Namely, not only do we not know whether $\mathsf{NP} = \mathsf{UP} \Rightarrow \mathsf{PH}$ collapses, we don't even know whether this question requires nonrelativizing techniques. Does there exist an oracle relative to which $\mathsf{NP}=\mathsf{UP}$ but $\mathsf{PH}$ doesn't collapse to the second level? Or doesn't collapse at all?)

If we instead assume surjectivity, the assumption that all onto honest poly-time functions have poly-time inverses was studied by Fenner-Fortnow-Naik-Rogers. They dubbed it "Hypothesis Q", and showed it is equivalent to a number of other complexity-theoretic statements. One of these equivalent statements is: for all $S \subseteq SAT$ such that $S \in \mathsf{P}$, there is a polynomial-time computable $g$ such that for all $\varphi \in S$, $g(\varphi)$ outputs a satisfying assignment to $\varphi$. (Note that this does not follow automatically from the usual self-reducibility of SAT, since one does not know that the self-reduction takes formulas in $S$ to formulas in $S$.)

If instead of poly-time we talk about polynomial circuit size, Birget showed that such functions exist unless $\mathsf{PH} = \mathsf{\Sigma_2 P}$.

For one-way functions with additional properties (such as commutativity, associativity, and a few others), their existence is related to the existence of ordinary one-way functions, see Hemaspaandra-Rothe-Saxena.

honesty- that the length of the output must be nearly equal to some polynomial of the length of the input. As Joel's example shows, this requirement is essential... $\endgroup$