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 $\in \mathbb R$.

The notation
$$
f: [1,5] \rightarrow \mathbb{R}
$$
means that $f$ is a function whose domain is to taken to be the interval $[1,5]$ and whose codomain is $\mathbb{R}$ (i.e. all the outputs of $f$ fall into $\mathbb{R}$). It makes no claims about surjectivity or injectivity; you must analyze the function itself to decide that.
To address your example, you define
$$
f: [1,5] \rightarrow ?
$$
$$
f(x) = x^2,
$$
where the question mark means we aren't sure what to put there yet. Since since the square of any number in the interval $[1,5]$ is a real number, then indeed $\mathbb{R}$ is an acceptable codomain for $f$, so we could write
$$
f: [1,5] \rightarrow \mathbb{R}.
$$
Notice that, as we have defined it, $f$ is not surjective, since some numbers in $\mathbb{R}$ cannot be reached by $f$ (like 36).
Looking a little closer, we might notice that the square of any number in the interval $[1,5]$ falls in the interval $[1,25]$. Thus, we would also be correct in writing
$$
f: [1,5] \rightarrow [1,25].
$$
After this change in codomain, $f$ becomes surjective, since every number in $[1,25]$ can be reached by $f$.
