Describe the free algebra on one generator in the variety of all algebras with two unary operations $f, g$. Do the same for the subvariety axiomatized by $f(g(x)) = g(f(x))$.
I'm not sure how to get started with this. Any insight would be greatly appreciated.
I believe this hint will give you insight enough; otherwise, I can expand this to a complete answer.
Let $\mathcal B$ be the variety of bi-unary algebras (say, with operations $f$ and $g$)
and $\mathcal V$ be its sub-variety given by the identity $fg(x)=gf(x)$.
The free algebra on one generator, say $1$, over $\mathcal B$ is represented by an infinite binary tree (say $f$ corresponds to the left hand branch, from each node, and $g$ to the right one);
The free algebra on one generator, say $(0,0)$, over $\mathcal V$ is isomorphic to $(\mathbb N, \sigma)^2$ (where $\sigma$ is the successor operation), where $f$ corresponds to increments on the first coordinate and $g$ on the second.
Prove each of these by checking these algebras have the universal mapping property for the correspondent varieties over a singleton.