Is a set of values of an exponential function uncountable?

Question

E.G. $f(x) = 2^x$

If yes, is there an easy, informal proof that a layman could understand?

Answer 1

Suppose the function is defined on some domain $D \subseteq \Bbb R$. The function $f(x)=2^x$ has an inverse on $(0,\infty)\supseteq f[D]$, namely $y \to {^2}\log(y)$, so that $f$ is injective. By standard set theory this means that $|D|\le |f[D]|$, i.e. the image set is at least as large as the domain in cardinality. So yes if your domain is uncountable.