Meaning of $f:[1,5]\to\mathbb R$ I know $f:[1,5]\to\mathbb R$, means $f$ is a function from $[1,5]$ to $\mathbb R$. I am just abit unclear now on the exact interpretation of "to $\mathbb R$". Is $1\le x\le 5$ the domain? And is $\mathbb R$ the co-domain (or image?)?
Is my interpretation in words ---$f$ is a function which takes a number $1\le x\le 5$, and maps it onto the real numbers $\mathbb R$, correct?
Suppose we take $f=x^2$ and $x=2$, does $f:[1,5]\to\mathbb R$ hold? So thus the function gives us $4$, which is $\in \mathbb R$.

It's correct if you say "into" rather than "onto".  The domain is $[1,5]=\{x: 1\le x\le 5\}$.  For every number $x$ in the domain, $f(x)$ is in $\mathbb{R}$.
"Onto" is often taken to mean "surjective" (except that "onto" is also used as a preposition).  I.e. if $f$ is said to map the set $[0,5]$ "into" $\mathbb{R}$, that means only that for every number $x$ in the domain, $f(x)$ is in $\mathbb{R}$, but if $f$ is said to map the set $[0,5]$ "onto" $\mathbb{R}$, that means that in addition to the fact that for every number $x$ in the domain, $f(x)$ is in $\mathbb{R}$, it is also the case that for every member $y$ of $\mathbb{R}$, there is some number $x$ in $[1,5]$ such that $f(x)=y$.  So, for example $f(x)=x^2$ maps $[0,1]$ into, but not onto, $\mathbb{R}$.  It's not onto because for example $26\in\mathbb{R}$ but there is no number $x\in[1,5]$ such that $f(x)=26$.
