calculus - Function that is uniformly continuous but not bounded?

I've been given the following question but I'm unsure if there are actually any answers:

Give examples of functions $f,g: \mathbb{R}\to\mathbb{R}$ which are uniformly continuous such that $f$ is not bounded but $g$ is bounded.

I know that if $f:(a,b)\to\mathbb{R}$ is uniformly continuous then $f$ is bounded so surely there is no such example for $f$ ?

Answer

What about $f(x)=x$ and $g(x)=A$?