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 $\in \mathbb R$.

Your function $f$ can be defined on all sorts of domains. You can define it on [2,2] (i.e. at the point 2) or on the closed interval [1,5] (real numbers from 1 to 5 including 1 and 5). You can define it on subsets of the Real or Complex numbers. But if you are worried about such things with such functions you will give yourself a headache.
You could also (for completeness) define it on an interval like (1,5) - real numbers between 1 and 5 but not including 1 or 5.
Whichever domain of definition you were to choose, the vale at the Real number 2 would be the same (or at 2 identified as a complex number on the Real axis).
