Question concerning the domain of a function

Question

In the function $f:\mathbb{N}\rightarrow\mathbb{N}$ defined by $f(x)= 5x$ why is range not equal to co domain? I don't understand why this is not a surjective function while the same function is surjective if $f:\mathbb{R}\rightarrow\mathbb{R}$.

Answer 1

If $f$ is defined on the natural numbers you have $im(f)=5\mathbb{N}=\{5,10,15,20,\dots\}$. If $f$ is defined on $\mathbb{R}$ then given any $x\in \mathbb{R}$ there is a $y:=\frac{x}{5}\in\mathbb{R}$ such that $f(y)=x$. Hence $f$ is surjective.