Describe an injective function from the real numbers to the real numbers that is not a bijection

Question

Wouldn't the set of real numbers automatically biject the set of real numbers since they are the same set of numbers?

Answer 1

Consider the function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=2^x$. This function is injective because if $2^x=2^y$, then $$ x= \log_2(2^x)=\log_2(2^y)=y. $$ However $f$ is not a surjection, because $f(x)>0$ for every $x\in\mathbb{R}$.