Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy:

$$
f(x)^{3}+f(x) \leq x \leq f\left(x^{3}+x\right)
$$
This exercise is not very difficult, but it has been deliberately messed up. Suppose the equation was

$$
f(x)^{3} \leq x \leq f\left(x^{3}\right)
$$

then the first inequality indicates that

$$
f(x)^{3} \leq x \text { so } f(x) \leq \sqrt[3]{x}
$$

and if we look at the second inequality with $y=\sqrt[3]{x}$, we get

$$
\sqrt[3]{x} \leq f\left(\sqrt[3]{x}^{3}\right)=f(x)
$$

. However, here we need to look a bit more closely at the function $g: \mathbb{R} \rightarrow \mathbb{R}, g(x)=x^{3}+x$. This function is strictly increasing and continuous on $\mathbb{R}$ and $g(\mathbb{R})=\mathbb{R}$, so $g$ is a bijection from $\mathbb{R}$
to $\mathbb{R}$. We denote $g^{-1}$ as its inverse function. We can now use the same technique as before: the first inequality shows that $f(x) \leq g^{-1}(x)$ and the second inequality taken at $g^{-1}(x)$ that $g^{-1}(x) \leq f(x)$.

Conversely, $g^{-1}$ is a solution to the equation.