What is Function: Definition and 1000 Discussions

In mathematics, a function is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. It is customarily denoted by letters such as f, g and h.If the function is called f, this relation is denoted by y = f (x) (which reads "f of x"), where the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (e.g., f is a function of the variable x).A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means of illustrating the function.
Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.

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  1. docnet

    Please help me understand a definition of a covariance function

    On page 3 of the lecture notes for Stochastic Analysis, it says '##B(s,t)## is the covariance function ##\mathbb{E}[X_sX_t]-\mathbb{E}[X_s]\mathbb{E}[X_t]##. Then On page 5, it says the notes also say that 'the covariance function ##B(s,t)## of a strongly stationary stochastic process is...
  2. CECE2

    I Can a function inside the integral be erased?

    Given that $$\int_a^b f(x)g(x) \, dx = \int_a^b f(x)h(x) \, dx$$ and $$f(x)=e^x$$, is it true that $$\int_a^b g(x) \, dx = \int_a^b h(x) \, dx$$?
  3. CECE2

    A Can a function inside the integral be erased?

    Given that $$\int_a^b f(x)g(x) \, dx = \int_a^b f(x)h(x) \, dx$$ and $$f(x)=e^x$$, is it true that $$\int_a^b g(x) \, dx = \int_a^b h(x) \, dx$$?
  4. flyusx

    Is a Distribution Function a Ratio of Differentials?

    I read on a post here titled 'Understanding Fourier Transform for Wavefunction Representation in K Space' that if one represents the squared-amplitude as a ratio of differentials, the solution is given. Letting the squared-amplitude be ##\phi##. $$\frac{d\phi}{dp}=\frac{d\phi}{dk}\frac{dk}{dp}$$...
  5. chwala

    Solve the problem involving the cubic function

    The problem and solution are posted... no. 8 I may need insight on common difference ... In my lines i have, Let the roots be ##(b), (b-1)## and ##(b+1)##. Then, ##x^3-3bx^2+3cx-d = a(x-b(x-b+1)(x-b-1)## ##x^3-3bx^2+3cx-d= a(x^3-3bx^2+3b^2x-x-b^3+b)## ##a=1##. Let...
  6. chwala

    Show that the given function is continuous

    Refreshing... going through the literature i may need your indulgence or direction where required. ...of course i am still studying on the proofs of continuity...the limits and epsilons... in reference to continuity of functions... From my reading, A complex valued function is continous if and...
  7. MatinSAR

    Find force as a function of position: F=F(x) using v=v(t)

    If we consider ##v=-3t^2## then: $$x=-t^3$$$$a=-6t$$ Using ##t=-x^{1/3}## we have : ##a=-6(-x^{1/3})=6x^{1/3}##. My answer suggust that ##F=Ax^{1/3}## but in options we have ##F=-Ax^{1/3}##. Can someone guide me where my mistake is?
  8. brotherbobby

    State the domain and range for a given function

    Attempt : Let me copy and paste the problem as it appeared in the text. Please note that the given problem appears in part (b), which I have underlined in red ink in this way ##\color{red}{\rule{50pt}{1pt}}## Clearly the domain is ##\boxed{\mathscr{D}\{f(x)\}...
  9. brotherbobby

    B Understanding the Relationship Between a Function and Its Inverse

    I could only verify this for a few elementary functions. Does a proof exist? Does it go beyond the realms of high school mathematics? Many thanks.
  10. A

    How to find the range of a function with square roots?

    $$y = f(x) = \sqrt{9-x^2}$$ According to me, Domain: $$ 9-x^2 \geq 0 \implies (x+3)(x-3) \leq 0 \implies x \in [-3,3] $$ which is correct, but this is how I calculate the range: $$y = \sqrt{9-x^2} \implies y^2 = 9-x^2 \implies x^2 = 9-y^2$$ Now, since $$ 9-x^2 \geq 0 $$ we get $$9-9+y^2 \geq 0...
  11. chwala

    Find the domain and the range of ##f-3g##

    Am refreshing on this, For the domain my approach is as follows, ##(f-3g)x = f(x)-3g(x)## ##=x-3-3\sqrt{x}##. The domain of ##f-3g## is given by ##f∩g = [{x: x ≥0}]## We have ##y= x-3-3\sqrt{x}=(\sqrt x-\frac{3}{2})^2-\dfrac{21}{4}##. The least value is given by...
  12. S123456

    I Local inverse of non bijective functions

    Hi, I am having a hard time trying to solve this question. How do I find the local inverse at x0? f (x) = x^4 − 4x^2 Find an expression for f^−1 for f at the point x = −2. Thanks a lot! I would really appreciate any help!!
  13. C

    I I need to write a function for DPI screen scaling with parameters

    i need to write a function for DPI screen scaling, so the parameters is from 100 (percentage) to 350 (percentage) and increases at 25 (percentage) increase, it will subtract additional 1 DPI so for example: 100% = 96 DPI which is -4 125% =120 DPI which is -5 150% =144 DPI which is -6 175%...
  14. brotherbobby

    To find the range of a given ##\sin## function

    Attempt : The domain of the function ##\sin(3x^2+1)## is clearly ##x\in (-\infty, +\infty)##. The values of ##x## go into all quadrants where the ##\sin## curve is positive and negative. Hence the range of the function ...
  15. P

    Injective Function: Proving Correctness with Singlets ##S## and ##T## in ##X##

    I operated by placing ##S## and ##T## to two singlets belonging to ##X## and therefore established that for ##T, S \in X##, therefore ##f (T) = f (S) \implies S = T##, consequentially: $$f (T \cap S ) = f (T \cap T) = f (T) \cap f (T) = f (T) \cap f (S)$$. I would like to know if my procedure is...
  16. neilparker62

    I Question on Lambert W function

    In the following I ask WA to solve the given equation and it produces a solution using the Lambert W function. I thought : $$W(x*e^x) = x$$ but here it seems $$W_n \left(\frac{-MT}{P}*e^{\frac{-MT}{P}}\right) \neq \frac{-MT}{P}$$ Is there a difference between ##W(x)## and ##W_n(x)## ?
  17. al4n

    B Looking for a specific periodic function

    Is there a function that outputs a 1 when the input is a multiple of a number of your choice and 0 if otherwise. The input is also restricted to natural numbers. The only thing I can come up with is something of the form: f(x) = [sin(ax)+1]/2 but this does not output a 0 when I want it.
  18. Euge

    POTW A Function in the Continuous Hölder Class

    Let ##0 < \alpha < 1##. Find a necessary and sufficient condition for the function ##f : [0,1] \to \mathbb{R}##, ##f(x) = \sqrt{x}##, to belong to the class ##C^{0,\alpha}([0,1])##.
  19. B

    Partial Derivative Simplification

    Hi there! I would like to know if the following simplification is correct or not: Let A be a function of x, y, and z $$\frac{\partial^2A}{\partial x^2}+\frac{\partial^2A}{\partial y^2}$$ $$=\ \frac{\partial^2A\partial y^2+\partial^2A\partial x^2}{\partial x^2\partial y^2}$$...
  20. SaschaSIGI

    I Understanding Hessian for multidimensional function

    Hello everybody, I have a question regarding this visualization of a multidimensional function. Given f(u, v) = e^{−cu} sin(u) sin(v). Im confused why the maximas/minimas have half positive Trace and half negative Trace. I thought because its maxima it only has to be negative. 3D vis 2D...
  21. chwala

    Find the derivative of the given function

    Let's see how messy it gets... ##\dfrac{dy}{dx}=\dfrac{(1-10x)(\sqrt{x^2+2})5x^4 -(x^5)(-10)(\sqrt{x^2+2})-x^5(1-10x)\frac{1}{2}(x^2+2)^{-\frac{1}{2}}2x}{[(1-10x)(\sqrt{x^2+2})]^2}##...
  22. I

    Show that the given function is decreasing

    As a follow up for : https://www.physicsforums.com/threads/let-k-n-show-that-there-is-i-n-s-t-1-1-k-i-1-2-k-i-1-4.1054669/ show that ## \alpha\left(k\right)\ :=\ \left(1-\tfrac{1}{k}\right)^{\ln\left(2\right)k}-\left(1-\tfrac{2}{k}\right)^{\ln\left(2\right)k} ## is decreasing for ##...
  23. Silvia2023

    For this Partial Derivative -- Why are different results obtained?

    Given a function F(x,y)=A*x*x*y, calculate dF(x,y)/d(1/x), to calculate this derivative I make a change of variable, let u=1/x, then the function becomes F(u,y)=A*(1/u*u)*y, calculating the derivative with respect to u, we have dF/du=-2*A*y*(1/(u*u *u)) replacing we have dF/d(1/x)=-2*A*x*x*x*y...
  24. PeaceMartian

    I What are the Zeta Function and the Riemann Hypothesis?

    What is the zeta function and the Riemann hypothesis.
  25. phos19

    I Fermi energy for a Fermion gas with a multiplicity function ##g_n##

    I ran across the following problem : Statement: Consider a gas of ## N ## fermions and suppose that each energy level ## \varepsilon_n## has a multiplicity of ## g_n = (n+1)^2 ##. What is the Fermi energy and the average energy of this gas when ## N \rightarrow \infty## ? My attempt: The...
  26. E

    Vector is a function of its position or not?

    At first I thought that this force vector ## \vec F = 3 \hat x + 2 \hat y ## is a function of ## x ## and ## y ##, which is to say that its magnitude and direction vary with the x and y positions, but this is not so, right? It's just a force with a constant magnitude and direction. And I can...
  27. MatinSAR

    Distance as a function of time for two falling stones

    I am aware that this question is very simple and basic. Using ##y(t)=y_0+v_{0,y}t-\frac {1}{2}gt^2## we can find distance as a function of time: ##|y_1-y_2|=|y_0+v_{0,y}t|=-y_0- v_{0,y}t## I assumed the downward direction to be negative. So as I wrote ##D(t)=-y_0- v_{0,y}t##. It tells that the...
  28. E

    Details regarding the high temperature limit of the partition function

    My main question here is about how we actually justify, hopefully fairly rigorously, the steps leading towards converting the sum to an integral. My work is below: If we consider the canonical ensemble then, after tracing over the corresponding exponential we get: $$Z = \sum_{n=0}^\infty...
  29. Mohmmad Maaitah

    How to find intervals where this function is decreasing and increasing?

    Please walk me step by step on how to do it (we don't have imaginary numbers so don't bring that up) Also how to put signs on the numbers line when I get minus in the root? (non solveable equation) Sorry for my English.
  30. C

    Finding where this function is increasing or decreasing

    For this, I first try to work out where function is increasing My working is ##f'(x) = 12x^3 - 12x^2 - 24x## For increasing, ##12x(x^2 - x - 2) > 0## ##12x > 0## and ##(x - 2)(x + 1) > 0## ##x > 0## and ##x > 2## and ##x > -1## However, how do I combine those facts into a single domain...
  31. C

    Graphing a piecewise function (Python)

    I am trying to write a python script to plot the function, Where ##V_0 = 5~V## ##t_0 = 10~ms## ##\tau = 5~ms## My script that I have written to try to do this is, Which plots, However, the plot is meant to look like this with the horizontal line. Can someone please give me some guidance to...
  32. L

    Discharging a capacitor -- Calculate the current as a function of time

    Hi, I am not sure if I have calculated the task b correctly. I always interpret an open switch as an infinitely large resistor, which is why no current is flowing through this "resistor". So there is no current in the red circle, as it was the case in task part a, but only in the blue circle...
  33. F

    I Partial derivatives of the function f(x,y)

    Hello, Given a function like ##z= 3x^2 +2y##, the partial derivative of z w.r.t. x is equal to: $$\frac {\partial z}{\partial x} = 6x$$ Let's consider the point ##(3,2)##. If we sat on top of the point ##(3,2)## and looked straight in the positive x-direction, the slope The slope would be...
  34. T

    I Infinite product representation of Bessel's function of the 2nd kind

    An infinite product representation of Bessel's function of the first kind is: $$J_\alpha(z) =\frac{(z/2)^\alpha}{\Gamma(\alpha+1)}\prod_{n=1}^\infty(1-\frac{z^2}{j_{n,\alpha}^2})$$ Here, the ##j_{n,\alpha}## are the various roots of the Bessel functions of the first kind. I found this...
  35. barryj

    I Formula for credit card balance as a function of payments

    I have been trying to find the financial formula that will give the balance of a credit card debt as a function of time. Example, at 18% interest, if I pay $150 a month how long will it take me to pay off my debt. When I google, I get pointers to Excello functions. I want to know the exact formula.
  36. A

    Oven controller block diagram, transfer function and temperature calcs

    FIGURE 5 shows an electrically heated oven and its associated control circuitry. The current, I, to the oven's heating element is fed from a voltage-controlled power amplifier such that I = EK1. A voltage, VD, derived from a potentiometer, sets the desired oven temperature, TD. The oven...
  37. kakaho345

    Finding free electron gas Green function in Fourier space

    As in title: Plugging in the definition is straight forward, I am too lazy to type, I will just quote the book Fetter 1971: Up to here everything is very straight forward, in particular, since we are working on free electron gas, ##E=\hbar \omega## However, I have no idea how to arrive...
  38. J

    I Solution of delayed forcing function

    Tried to figure out myself but have now admitted defeat, requesting some guidance from you good people. Not looking for any specific answers, unless the problem is my working out and not my process. If we take the following differential equation: ##y(t)'' + 4y(t) = 7u(t-2)## and determine...
  39. P

    Equation involving inverse trigonometric function

    I came across the mentioned equation aftet doing a integral for an area related problem.Doing the maclaurin series expansion for the inverse sine function,I considered the first two terms(as the latter terms involved higher power of the argument divided by factorial of higher numbers),doing so...
  40. ergospherical

    Speed of function vs lambda calls in python

    An observation I made earlier- something like def f(...): ... return ... def g: ... = f(...) was quite a bit slower than doing def g: f = lambda ... : ... ... = f(...) any reasons why?
  41. H

    I Symmetry and two electron wave function

    In the picture below we have two identical orbitals A and B and the system has left-right symmetry. I use the notation ##|n_{A \uparrow}, n_{A \downarrow},n_{B \uparrow},n_{B \downarrow}>## which for example ##n_{A \uparrow}## indicates the number of spin-up electrons in the orbital A. I would...
  42. C

    Showing piece-wise function continuous

    For this, , The solution is, However, should they not write ##f(x) = \cos x## on ##[\frac{pi}{4}, \infty)## Many thanks!
  43. C

    Using continuity to evaluate a limit of a composite function

    For this problem, The solution is, However, I tried to solve this problem using my Graphics Calculator instead of completing the square. I got the zeros of ##x^2 - 2x - 4## to be ##x_1 = 3.236## and ##x_2 = -1.236## Therefore ##x_1 ≥ 3.236## and ##x_2 ≥ -1.236## Since ##x_1 > x_2## then...
  44. C

    Interchanging x and y for inverse function

    For this, Why are we allowed to interchange x and y? Is it because the equation will still be true? Many thanks!
  45. Euge

    POTW Local Integrability of a Maximal Function

    Let ##f## be a measurable function supported on some ball ##B = B(x,\rho)\subset \mathbb{R}^n##. Show that if ##f \cdot \log(2 + |f|) ## is integrable over ##B##, then the same is true for the Hardy-Littlewood maximal function ##Mf : y \mapsto \sup_{0 < r < \infty}|B(y,r)|^{-1} \int_{B(y,r)}...
  46. chwala

    Write the given hyperbolic function as simply as possible

    My take; ##2\cosh x = e^x +e^{-x}## I noted that i could multiply both sides by ##e^x## i.e ##e^x⋅2\cosh x = e^x(e^x +e^{-x})## ##e^x⋅2\cosh x = e^{2x}+1## thus, ##\dfrac{e^x}{1+e^{2x}}=\dfrac{\cosh x + \sinh x}{e^x⋅2\cosh x}## ##= \dfrac{\cosh x +...
  47. C

    Is ##f(x)=2^{x}-1## considered an exponential function?

    I wonder if it's ##f(x)=2^{x}-1## considered an exponential function because in my textbook it's stated that the set of values of an exponential function is a set of positive real numbers, while when graphing this function I get values(y line) that are not positive(graph in attachments), so I am...
  48. ananonanunes

    Find limit of multi variable function

    This is what I did: $$\lim_ {(x,y) \rightarrow (1,0)} {\frac {g(x)(x-1)^2y}{2(x-1)^4+y^2}}=\lim_ {(x,y) \rightarrow (1,0)} {g(x)y\frac {(x-1)^2}{2(x-1)^4+y^2}}$$ I know that ##\lim_ {(x,y) \rightarrow (1,0)} {g(x)y}=0## and that ##\frac {(x-1)^2}{2(x-1)^4+y^2}## is limited because ##0\leq...
  49. Like Tony Stark

    Mixed states and total wave function for three-Fermion-systems

    I've already calculated the total spin of the system in the addition basis: ##\ket{1 \frac{3}{2} \frac{3}{2}}; \ket{1 \frac{3}{2} \frac{-3}{2}}; \ket{1 \frac{3}{2} \frac{1}{2}}; \ket{1 \frac{3}{2} \frac{-3}{2}}; \ket{0 \frac{1}{2} \frac{1}{2}}; \ket{0 \frac{1}{2} \frac{-1}{2}}; \ket{1...
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