What does $f:[a,b] \to [a,b]$ mean?

Question

Does $f:[a,b] \to [a,b]$ mean that $f(a)=a$, or does it mean that $f(a)<a$ or $f(a)>a$? If there there any values between $[a,b]$, then how is the $f(a)$ or the $f(b)$ related to these values?

Answer 1

Once you get to calculus and linear algebra (functional analysis too), we may look at general structures of functions just by their domains and their codomains. To say $f: S \to T$ is to say that $f$ maps elements from the set $S$ into the set $T$. For example,

$$ f: [a, b] \to \mathbb{R}$$

is a function that takes values from the interval $[a, b]$ to the real numbers. It doesn't tell you about individual values, just where these images live.