Why is a function thought of as a rule rather than just a formula?

A function is a rule that gives exactly one output for each input, and a formula is only one way of writing that rule down. The formula is a recipe; the function is the assignment itself, which may also be given by a table, a procedure, or a measured record.

Read the arrow as becomes. The rule called $f$ takes whatever number arrives as $x$ and returns twice that number plus three. The letter $x$ is a placeholder, not a fixed value: renaming it throughout, say to $t$, gives $f : t \mapsto 2t+3$, which is the same rule. The expression $2x+3$ is only the recipe for the answer. Two different recipes can produce the same rule, and one recipe can be read in more than one way unless the permitted inputs are stated.

Hand the rule four numbers and the arithmetic is ordinary. At $0$ it doubles to $0$ and adds three to give $3$. At $1$ it doubles to $2$ and adds three to give $5$. At $2$ it doubles to $4$ and adds three to give $7$. At $-2$ it doubles to $-4$ and adds three to give $-1$. Each extra one on the way in adds two on the way out: from $0$ to $1$ the answer rises from $3$ to $5$, and from $1$ to $2$ it rises from $5$ to $7$. That steadiness belongs to this rule, not to the letters used to write it. The same rule can be written as $3+2x$, or as $2(x+1)+1$, or as a two column table of inputs and answers. Changing the recipe does not change the function if every permitted input still gets the same single answer.

The natural guess is that if you have a formula, you have the function. That fails because a formula is a compressed instruction, and it can be ambiguous until the rule says what may be handed in and what is done with it. The formula $1/x$ does not by itself tell you whether $0$ is allowed; the rule must exclude $0$, or the assignment is not defined there. A formula can also be rearranged without changing the function, which shows that the function was never the particular string of symbols. And a formula can be incomplete: writing $x^2$ does not say whether the inputs are all real numbers, all integers, or only the numbers between $0$ and $1$. The rule is the thing that answers exactly once for each permitted input.

Check the agreement by hand. At $x=0$, $g(0)=0+0.3\sin0=0$. At $x=1$, $g(1)=1+0.3\sin\pi=1$. At $x=2$, $g(2)=2+0.3\sin2\pi=2$. At $x=3$, $g(3)=3+0.3\sin3\pi=3$, and at $x=4$, $g(4)=4+0.3\sin4\pi=4$. The first rule returns $0,1,2,3,4$ at those inputs, exactly the same list. Now check halfway between $0$ and $1$: $f(1/2)=1/2=0.5$, while $g(1/2)=1/2+0.3\sin(\pi/2)=0.5+0.3=0.8$. The five whole number answers did not determine the rule, because both rules produced them. No finite list of checked inputs can separate every possible pair of rules either: a third rule can be built to agree at any larger set of points you name.