Right now I am in front of these expressions
objective function
objective functionals
which seem to be very often used, but I've still not understood their meanings.
definitionlinear programmingterminology
Right now I am in front of these expressions
objective function
objective functionals
which seem to be very often used, but I've still not understood their meanings.
Basically, there are the following inclusions
$(\star)$ $\mathcal{D}(\mathbb{R}^n) \subset \mathcal{S}(\mathbb{R}^n) \subset \mathcal{E}(\mathbb{R}^n)$ and $\mathcal{E}'(\mathbb{R}^n) \subset \mathcal{S}'(\mathbb{R}^n) \subset \mathcal{D}'(\mathbb{R}^n)$.
Also $L^p(\mathbb{R}^n) \subset \mathcal{S}'(\mathbb{R}^n)$. Almost all of these inclusions are also continuing, i.e. $\mathcal{D}_K(\mathbb{R}^n) \hookrightarrow \mathcal{S}(\mathbb{R}^n)$, or also $L^p(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$, this means that the inclusion operator $\iota : L^p(\mathbb{R}^n) \longrightarrow \mathcal{S}'(\mathbb{R}^n)$ is continuous with respect to topology defined in those spaces, which basically may be a topology induced by a separable seminorms family or by norm. Where
(a) $\mathcal{D}(\mathbb{R}^n)$ is the space test functions
(b) $\mathcal{S}(\mathbb{R}^n)$ is the Schartz space
(c) $\mathcal{E}(\mathbb{R}^n)$ is the space of the regular functions
(d) $\mathcal{E}'(\mathbb{R}^n)$ is the space of the distribution with compact support
(e) $\mathcal{S}'(\mathbb{R}^n)$ is the space of the tempered distributions
(f) $\mathcal{D}'(\mathbb{R}^n)$ is the space of the distributions
For example, we have the inclusions $\mathcal{E}'(\mathbb{R}^n) \subset \mathcal{S}'(\mathbb{R}^n) \subset \mathcal{D}'(\mathbb{R}^n)$ because the inclusions $\mathcal{D}(\mathbb{R}^n) \hookrightarrow \mathcal{S}(\mathbb{R}^n) \hookrightarrow \mathcal{E}(\mathbb{R}^n)$ are continuous and dense with respect to topology of these spaces and then, for example, the application $v \in \mathcal{E}'(\mathbb{R}^n) \longrightarrow v=u_{\mathcal{D}(\mathbb{R}^n)} \in \mathcal{D}'(\mathbb{R}^n)$ is linear and one-to-one. Therefore each distribution determines a continuous linear functional $v : \mathcal{E}(\mathbb{R}^n) \longrightarrow \mathbb{C}$ (with respect to convergence in $\mathcal{E}(\mathbb{R}^n))$ which it is a compact support distribution, likewise $v : \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathbb{C}$ is a temperate distribution.
Look, here are a lot of theorems to prove, I'm doing you a summary, and basically $\mathcal{S}'(\mathbb{R}^n)$, that is the space of tempered distributions, it's a good space to define the Fourier transform for duality, because essentially the Fourier transform of Schwartz functions satisfies the very useful properties.
Most of these I have studied in several books, there is not just one. "Real Analysis: Modern Techniques and Their Applications" by Folland and "Linear Functinoal Analysis" by "J. Cerda" cover a large part of these topics.
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:
A function is a machine that takes in numbers and spits out other numbers. A formula is a mathematical expression that I can use to calculate. Functions are more general objects than formulas, because not every function can be written as a formula. My favorite example is the floor function, which works like this: it takes in a number, cuts off the decimal, and spits out the whole number that's left over. [examples]. This function makes perfect sense, even though I can't write down a "formula" for it like x^2 or $\sin(x)$.
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
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]...
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]:
The simplest explanation is in the answer to the linked post: "A formula is a string of symbols, arranged according to mathematical grammar [i.e. an expression of the mathematical language]. A function is a mathematical object." The clear distinction is that between the world of (mathematical) objects and the language used to speak of them (exactly like natural language). –
Best Answer
The concept of $objective$ $function$ is used in optimization. It denotes the function you want to minimize or maximize. It is used in many areas as in curve fit, regression, economics, ...
If there is any relevant area you would like to discuss, just post.
Just by the way, I was born in Dar El Beida (normal name hidden !).