I hope this question fits in this site. I was asked by my student when I was explaining the concept of relation and function and after thinking, I still can't differentiate those two concepts (formula and function). I've looked for the answers from other sites and I have read these:
The first link above tells us
'The difference between a formula and function is that a formula is defined as the statement used for the calculation [a]. These formulas
could be simple or complex and always start with equal to operator [b].
While function is defined as the code that is designed for the
calculations and is used inside the formula [c].'
We know a function can be used to calculate to find the result of the calculation from the range with the given number taken from the domain [a]. For instance, the exponential function
$$P(t) = P_0e^{kt}$$
can be used to calculate the population in the region X.
A function also has the equal sign, because if it doesn't, I'd say it's an expression [b] & [c]. And I don't know about the relevancy of the second link since the context is used in Excel. But it says that
Formula(s) is/are nothing but doing some arithmetic operation(s) or
calculation(s).
Does that mean these are formulas?:
$$2+2=4\quad 5-3=2\quad 2(3+5)/4=4$$
How to tell to my students about their difference? Are the exactly the same concept? I have a class tomorrow and I've promised to answer their question that day.
Best Answer
In my estimation, you are asking two different questions here. The one about what might tell your students is easier to answer, but there are lots of different answers that are reasonable. Here is what I would say to, e.g., a college student in any course before linear algebra:
None of the first three sentences are true (the fourth one is!), even to the extent that they are meaningful. But the lies are mild, and they help simplify the language to get the point across.
The second question is what the difference actually is. I will not come to any forceful conclusion on this question, but instead frame my thoughts as a series of remarks.
Remark 1. This question has been discussed a lot on MSE, because, as you have observed, it is subtle, and it concerns relatively elementary objects. Exactly which question it is a duplicate of is not totally clear, though.
Remark 2. I can't open the first link, but based on the language used, it kind of sounds like this source is using the programming definition of a 'function', which differs in a subtle way from the mathematical definition. In particular, we usually do not agree with [c]: for us, the function is the encapsulating object, and there are various methods by which a function's values may be given, one of which is a formula (see this Wikipedia quote)
Remark 3. Continuing on this, you write
I know no such thing. A function may encode a calculation (or it may not!) but it is odd to me to say that we use a function to calculate elements in its range. For instance, in your example I would not say I am using the function $P$ to calculate $P(3)$; I guess I can see why you would say this, but it doesn't reflect how the calculation actually proceeds. When I am actually getting around to doing the computation, what I am using is the expression $P_0e^{kt}$ by substituting $t=3$. The result of this calculation is certainly in the range of $P$; this is guaranteed by the equation $P(t)=P_0e^{kt}$... indeed this equation is what convinced me to use the expression to compute $P(3)$.
Remark 4. The word "formula" did not appear in the previous remark, what's up with that? Well, in casual speech, I would describe both $P(t)=P_0e^{kt}$ and $P_0e^{kt}$ as formulas for $P(t)$. If I am required to be more careful, I would not use the word at all.
Remark 5. Logicians have strong opinions about [read: a definition of] what a "formula" is, that does not align well with the common-language meaning, despite being related. (The tension is well-illustrated by this answer.) To answer your question, the three equalities in your question undoubtedly are formulas if we are using the logician's definition. Still, I would not call them formulas because for me a formula needs to be something that I can compute with; those equalities look to me like a "record" of a computation that has already been completed.
As this answer has now been accepted, let me highlight Mauro Allegranza's very useful comment [emphasis mine]: