The Name of a Rule

A rule can be given a name. Function notation writes $f\left(x\right) = 2x+1$ to name the rule "double, then add 1" as $f$. Then $f\left(3\right)$ means the rule $f$ applied to the input 3. To work it out, substitute 3 for $x$ and compute:

$$f\left(3\right) = 2\cdot 3+1 = 7$$

You read $f\left(3\right)$ as "f of 3". It is a value, not a multiplication. $f\left(3\right)$ is the output the rule gives for the input 3, the $y$ that belongs to $x = 3$.

Different rules get different names. Take $f\left(x\right) = 2x+1$ and $g\left(x\right) = x-4$. The same input 3 gives $f\left(3\right) = 7$ and $g\left(3\right) = -1$. Each function uses its own rule.