Challenge 67: An Almost Symmetric Functional Equation

  • Published on Nov 15, 2018
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Comments • 66

  • Zicar
    Zicar 10 months ago

    I divided the first formula by xy (x and y can't be 0) and arrived at f(x) - f(y) = f(x) / y - f(y) / x - 2/y +2/x.
    then i plugged in y = 2018 and arrived at the formula f(x) - f(2018) = f(x) / 2018 - f(2018)/x - 2/2018 + 2/x.
    i derived everything and arrived at f'(x) = f'(x) / 2018 + f(2018)/x^2 - 2/x^2.
    then i plugged in x = 1 and arrived at 2018 = 1 + f(2018) - 2 f(2018)-1 = 2018 f(2018) = 2019

  • Yaman Sanghavi
    Yaman Sanghavi 10 months ago

    Just put y=2 and rearrange to get f(x)
    And use f'(1) to find the unknown constant f(2) in the expression of f(x)

    PRAKHAR AGARWAL 10 months ago

    The answer os 2019
    Partial Differentiate the equation with respect to x to get a new functional equation. Put y=1 to get f(1)=2.
    Again partial different the new functional equation with respect to x. After canceling out the factor of (y-1) from the new differential equation, we get a differential equation in x.
    Which can be solved to get f(x) =2020-(2018/x)
    Put x=2018 to get 2019

  • miner Zhao
    miner Zhao 10 months ago +1

    just give x=1 (or y=1)we can get f(1)=2.then changing this equation form,we can get x.f(x)-2 over x-1 equal to y.f(y)-2 over well we derivate this to X,we get f(x)+x.f'(x)=y.f(y)-2/ let x=1, y.f(y)-2/y-1=f(1)+1.f'(1)=2020,after that let y=2018,so f(2018)=2019

  • Tamerlane
    Tamerlane 10 months ago

    Answer is 2020

    • Tamerlane
      Tamerlane 10 months ago

      Let's transform equation into the following:
      xf(x) (y-1) + 2x = yf(y) (x-1) + 2y
      Let's take y = 2:
      xf(x) +2x = 2f(2)(x-1)+4
      Therefore f(x) = (2f(2)-2) + (4-2f(2))/x
      But then
      f'(1)=2f(2)-4=2018 => f(2) = 1011.
      Now we know that
      f(x) = 2020-2018/x
      And therefore

  • Leonardo Costa
    Leonardo Costa 10 months ago

    Ans: 2019.
    By rewriting the equation you find that (yf(y)-2)/(y-1) is constant, which lets you write f(x) as a-b/x. Now by differentiating and plugging x=1 into the equation, you're able to solve for the constant and finally have f(x)=2020-2018/x. Meaning f(2018)=2019

  • Sébastien Louchart
    Sébastien Louchart 10 months ago +4

    Hello. I proved the continuity of f over its domain (x=t, y=a than lim(t->a) f(t) = f(a)). This done I felt more comfy to try to play with the functional identity. I proved f(1) = 2 simply plugging x=1 and y=2. I then derived the expression with respect to x to find f'(x) = -f(x)/x + yf(y)/y-1 * 1/x - 2/(y-1) * 1/x. evaluating for f'(1) = 2018 leads to f(y)= 2020(y-1)/y + 2/y hence f(2018) = 2019. I'm still working to prove derivability of f over its domain, though :/

    • Sébastien Louchart
      Sébastien Louchart 10 months ago

      The proposed function is defined over R* and it can be proved it is continuous over R*. For God's sake it's even differentiable over R* ;)

    • Sébastien Louchart
      Sébastien Louchart 10 months ago

      @Numer in that case. You're right. The function isn't continuous in 0.

    • Numer
      Numer 10 months ago

      @Sébastien Louchart I mean correct me, if i'm wrong, but just because the function is defined everywhere doesn't automatically imply, that the limit at every point exists, right? Take for example f: R->R defined by f(x) := sin(1/x) for x != 0 and f(0) := 0. Then lim_(x->0) f(x) doesn't exist, right?

    • Sébastien Louchart
      Sébastien Louchart 10 months ago

      @Numer the function f est defined over R*. You can thus take its limit anywhere on that domain

  • 025113322
    025113322 10 months ago

    The answer is 2019
    First, differentiate with respect to x with replacement of y=1 we get f(1)=2 then we replace x=1 we get yf(y) = 2020y-2018 so we plug in y=2018 we get f(2018) = 2019 as desired.

  • Elias Caeiro
    Elias Caeiro 10 months ago

    Let g(x) = f(x)*x. Our relation becomes yg(x) + g(y) + 2x = xg(y) + g(x) + 2y for all non-zero x, y. Let k = g(2)-2 g(2) = k+2.
    By plugging in y = 2, we get : 2g(x) + (k-2) + 4 = x(k-2) + g(x) + 2y g(x) = kx + 2 - k for all non-zero x.
    We have g'(x) = xf'(x) + f(x) so g'(1) = f'(1) + f(1) = f'(1) + g(1) = 2020. However, we also have g'(x) = k so g'(1) = k. Therefore k = 2020==> g(x) = 2020x - 2018.
    We get f(x) = (2020x-2018)/x = 2020 - 2018/x so f(2018) = 2019.

  • Brianchon
    Brianchon 10 months ago +3

    We can subtract 2 from both sides and rearrange to get (y-1)(xf(x)-2) = (x-1)(yf(y)-2); for x and both not equal to 1, this further rearranges to (xf(x)-2)/(x-1) = (yf(y)-2)/(y-1), so (xf(x)-2)/(x-1) is some constant C for all x other than 0 and 1. It follows that f(x) = C + (2-C)/x for all x other than 0 and 1. Since f is differentiable at 1, it is also continuous there, so our equation for f also holds at 1, and we can calculate the derivative: 2018 = f'(1) = C-2/(1)^2, so C = 2020. Thus, f(x) = 2020 - 2018/x for all x other than 0, and in particular, f(2018) = 2019.

  • iQuickdraw X
    iQuickdraw X 10 months ago

    Dividing x-y on both sides and take limit as y->x gives x^2*f'(x)+2=x*f'(x)+f(x), solving the differential equation with initial value f'(1)=2018 gives f(x)=2020-2018/x, so f(2018)=2019

    AKASH ASHOK 10 months ago

    The answer is 2019.
    Keeping y constant and differentiating wrt x,we get xyf'(x)+yf(x)-yf(y)+2=xf'(x)+f(x)
    Putting y=1,we get f(1)=2.
    Putting x=1,we get f(y)=2020-2018/y.
    Therefore f(2018)=2019.

  • Harald Schöpf
    Harald Schöpf 10 months ago

    rearrange to x*f(x)(y-1)+2x=y*f(y)(x-1)+2y -> f(z) must be of the form f(z)=a(z-1)/z+2 -> f´(z)=a/z^2 -> a=2018 ->2018*2017/2018+2=2019

  • Ashish kumar
    Ashish kumar 10 months ago

    Let a,b,c,d are roots of
    (X^2+7X+6)^2+7(X^2+7X+6)+6=X , given that a

    • Ashish kumar
      Ashish kumar 10 months ago

      @attyfarbuckle thank you , one question is f(f(x))=x ,but you take f(r)=r (why?)

    • attyfarbuckle
      attyfarbuckle 10 months ago +1

      Use f(x) =x^2 +7x +6.
      The strategy is to find r with f(r) = r.
      Any such r will also be a solution of the original equation.
      So solving x^2 + 7x + 6 = x gives x = -3 + sqrt(3) & x = -3 - sqrt(3)
      So now you've got 2 roots of the original equation, say r1 & r2.
      This means (X - r1) & (X - r2) are factors of (X^2+7X+6)^2+7(X^2+7X+6)+6 - X
      If you multiply out (X^2+7X+6)^2+7(X^2+7X+6)+6 - X = 0 then divide by (X^2 + 6X + 6) you get a quadratic which gives the other 2 roots:
      Let p(x) = x^2 + 6x + 6.
      Then (X^2+7X+6)^2+7(X^2+7X+6)+6 - X
      = (p(X)+X)^2 + 7(p(X)+X) + 6 - X
      = p(X)^2 + 2Xp(X) + X^2 + 7p(X) + 7X + 6 - X
      = p(X)[p(X) + 2X +7] + p(X)
      (it's just more algebra after this)

    • Ashish kumar
      Ashish kumar 10 months ago

      @Jackson Singh f(f(x))=x then next step ?
      Thanks Jackson

    • Ashish kumar
      Ashish kumar 10 months ago

      After f(f(x))=x , then ?
      Next step I can't understand about taking inverse

    • Jackson Singh
      Jackson Singh 10 months ago

      Lets the function be f(x) =x^2 +7x +6

  • Will Bishop
    Will Bishop 10 months ago

    Geez. Lots of algebra mistakes before I got it right but here we go.
    xy(f(x) - f(y)) + 2x = xf(x) - yf(y) + 2y
    xyf(x) - xf(x) - xyf(y) + yf(y) + 2x - 2y = 0
    (y-1)xf(x) - (x-1)yf(y) + 2x - 2y = 0
    Differentiate with respect to x:
    (y-1)(xf'(x) + f(x)) - yf(y) + 2 = 0
    Plug in x = 2018, y = 1:
    -f(1) + 2 = 0 --> f(1) = 2
    Plug in x = 1, y = 2018:
    2017(f'(1) + f(1)) - 2018f(2018) + 2 = 0
    2017(2018 + 2) - 2018f(2018) + 2 = 0
    2017*2020 + 2 = 2018f(2018)
    f(2018) = (2017*2020 + 2)/2018
    Let b = 2018 so we have f(b) = ((b-1)(b+2) + 2)/b = (b^2 + b - 2 + 2)/b = b + 1 = 2019
    So f(2018) = 2019.

  • A g
    A g 10 months ago

    Differentiate wrt x:
    y(f(x) -f(y)) + xyf'(x) +2 = f(x) + xf'(x)
    Let x = y
    (x^2-x)f'(x) = f(x)-2
    Solving the differential equation
    ln(f(x)-2) =ln(x-1/x) + c
    f'(1)=2018 so differentiating the above equation and solving it , we get c = 2018
    So f(x) -2 = 2018(x-1)/x
    f(2018) = 2017+2=2019.

  • NoName
    NoName 10 months ago

    Rearrange the equation:
    We can get rid of on EOF the functions by x=1 or y=1.
    Due to f'(1)=2018, f(1+dz)->2+2018dz
    Y=1+dz, X=2018
    We can drop the dz^2, as it is

  • Haradhan Datta
    Haradhan Datta 10 months ago

    Answer=2019 & f(X)=2020_(2018/X). By subtracting 2 from both sides and rearranging the functional equation,we have, (xf(X)_2)/(x_1)=(yf(y)_2)/(y_1) which is a constant for all values of X in domain of f. The given equation becomes an identity for X=y (in particular X=y=1).So we can take, (xf(X)_2)/(x_1)=k(constant) & hence f(X)=k_((k_2)/X) . Now f'(X)=(k_2)/x^2. But f'(1)=2018=(k_2), f(X)=2020_(2018/X). f(2018)=2019.

  • Jack Jones
    Jack Jones 10 months ago

    2019 is the answer
    put y=0 in the given equation we get x(f(x)-1)+2x=xf(x)-f(1)+2, from here we get f(1)=2
    Now, differentiate the equation keeping y as a constant we get
    Put x=1
    2y+f'(1)-yf(y)+2=2+f'(1) put the value of f'(1)=2018 and put y = 2018 we get the required answer f(2018)=2019

  • Jonty De Pledge
    Jonty De Pledge 10 months ago

    f(2018) = 2019
    First we substitute x=2 and y=1
    Rearranging gives us f(1)=2
    Since the expression is true for all non zero x and y we can differentiate it with respect to x treating y as a constant.
    Then substituting x=1, f(x)=2, f'(x)=2018 and rearranging gives f(y)=2020 - 2018/y.
    Substituting y=2018 we get f(2018)=2019

  • Kim Jong Un
    Kim Jong Un 10 months ago

    Take x=1
    Thus, for xyf(x)-xyf(y)+2x=xf(x)-yf(y)+2y we get
    isolating f(1) we get f(1)=2(y-1)/(y-1)=2
    so f(1)=2
    WLOG take the derivative of the expression w.r.t x and treating y as a constant you will get:
    plug in x=1 and substitute f'(1) for 2018
    Then, you get yf(1)+2018y-yf(y)+2=f(1)+2018
    Isolating f(y), you get yf(y)=2018y+yf(1)-f(1)-2016
    This leaves you with f(y)=2018+f(1)-(f(1)+2016)=2020-2018/y
    substitute y=2018 and you get f(2018)=2019

  • aby p
    aby p 10 months ago

    The solution to this is a little lengthy but I managed to get the ans so if we put x=1 in the functional equation we get f(1)=2 then if we put y=2 in the equation and then differentiate w.r.t x we get f(x)+x.f1(x)-2f(2)+2=0 here if we put f1(x) as dy/dx and put f(x) as y and then when we solve we get as 2020-(2018)/x and then when we put x=1 we get 2018 and then to get the ans we put x=2018 and we get the ans as f(2018)=2020-1=2019

  • Ankit Gupta
    Ankit Gupta 10 months ago

    Ans is 2019... F(x)=2020-(2018/x)

  • adandap
    adandap 10 months ago

    The answer is 2019. Rearranging the given equation we find that y(x-1)(f(x)-f(y))=(x-y)f(x)+2(y-x). Divide by (x-y) and take the limit as x-->y and you get a DE (yay!) y(y-1)f'(y)-f(y)=-2 Using the usual tricks (integrating factor, partial fractions) we solve to get f(y) = 2/y + (y-1)C/y (note that f(1)=2 for any finite C) So f'(y) = (C-2)/y^2 + a term proportional to (y-1). The condition on f'(y) gives us C=2. So f(2018) = (2020x2017+2)/2018 = 2019

  • Shakti Bhat
    Shakti Bhat 10 months ago

    The answer is 2019.
    First, substitute y=1 and x=2 to get f(1)=2.
    Now, put x=1 and solve for f(y) to get f(y) = (2020y-2018)/y.
    Now, substitute y=2018 to get f(2018) = 2019.

  • Mr L
    Mr L 10 months ago

    The answer is 2019. Divide by x-y, let y->x to obtain x(x-1) f'(x) = f(x) -2 ; a set of solutions is f(x) = 2/x + c(x-1)/x , away from 0, for any real c . Compute c using the condition f'(1) = 2018; c turns out to be 2020 and f(x) = 2020 - 2018/x , f(2018) = 2019. It remains to show that this is the only possible solution of the original problem

  • Rohit Agarwal
    Rohit Agarwal 10 months ago

    The answer is 2019. Take the partial derivative with respect to x on both sides. Plug in 1 for x and 2018 for y. You get that 2017*f'(1)=-2017f(1)+2018f(2018)-2. Next, plug in the same x and y for the original equation. The 2018*f(2018) cancels on both sides, and you get f(1)=2. Therefore, plugging this information in, you get into our partially solved one, you get f(2018)=2019.

  • Ricardo Mogollon
    Ricardo Mogollon 10 months ago

    the answer is 2019.
    Since the function in question is a single valued function, without loss of generality, we can take x as our variable and y as a positive real number, so the equation becomes more approachable, if we factorize the equation so that it becomes a more common notation of a function it becomes f(x)=(2(y-x)+yf(y)(x-1))/x(y-1), deriving the function and simplifying it becomes f'(x)=(y(f(y)-2)/((x^2)(b-1)) setting x as 1 and reordering we find out that y are inteconneted by the following relation. y=2018/(2020-f(y)), setting the previous relation on f(2018) we find out that f(2018)=(2019(f(b)-2))/(f(b)-2). Finally we can see that f(2018)=2019

  • Juan Pablo Gamucero
    Juan Pablo Gamucero 10 months ago

    Fix y, and derive from both sides, we get:
    From the second expression, if x=1
    In particular if y=1. We conclude f(1)=-2.
    So, our equation becomes
    And taking y=2018 we obtain:

  • Jackson Singh
    Jackson Singh 10 months ago

    The answer is 2019
    Firstly we know that the identuty is true for all x,y >0 so put x=1 and y=2 then we get f(1)=2 now we differentiate the identity to get and then put x=1 to get a equation in y and f(y) then we put y=2018 to get the answer 2019

    Mathias ICARTE MANCILLA 10 months ago

    I think that the answer is 2019. First take x=1 and y=2 for example, you will see that f(1)=2. Take the derivative with respect to x, which means that y is constant. Replace the x for 1 and solve for f(y) and that is 2020-2018/y. Thus f(2018) is 2019 :)

  • el tapa
    el tapa 10 months ago

    It's seems like you have yo use the CFT. Calculus fundamental theorem

  • Elias Caeiro
    Elias Caeiro 10 months ago +1

    Hi, I solved your problem but I had minor calculations mistakes… Is that a problem ? I replied to my first comment even though I shouldn't have…

    • Elias Caeiro
      Elias Caeiro 10 months ago

      @LetsSolveMathProblems Okay, thank you for responding.

    • LetsSolveMathProblems
      LetsSolveMathProblems  10 months ago +2

      Could you post an entirely new comment by copy-and-pasting your full solution? I almost always do not accept reply comments to reduce confusion on the order in which correct answers are submitted--Although USclip can sort regular comments in chronological order, it cannot do the same for replies.

  • GreenMeansGO
    GreenMeansGO 10 months ago

    What does f’ even mean in this context? How can we take the derivative if the function has inputs x AND y?

    • Cobalt314
      Cobalt314 10 months ago

      @GreenMeansGO You can implicitly differentiate by equating the differentials of both sides.

    • GreenMeansGO
      GreenMeansGO 10 months ago

      I understand but again, I am unable to find an expression for f’.

    • Jonty De Pledge
      Jonty De Pledge 10 months ago

      The function has only one input, this input could be either y or x.
      For example consider the function f(x)=x^2. The function f(y)=y^2 is an identical function we are just substituting in y instead of x. We could then write f'(x)=2x by differentiating with respect to x, or f'(y)=2y by differentiating with respect to y. Either way, we substitute in for x or y we get f'(1) = 1*2 = 2, it doesn't matter what variable we use.

    • GreenMeansGO
      GreenMeansGO 10 months ago

      Ok but I still don’t see how to derive an expression involving f’ due to there being two variables.

    • Cobalt314
      Cobalt314 10 months ago +1

      It's a function of 1 variable; f' is well-defined. It's just saying that for any two numbers you plug in (individually), the above relationship holds.

  • Parth Pawar
    Parth Pawar 10 months ago

    Answer is 2019.
    First perform partial differentiation on both sides with respect to x, you get:
    Now put y=x, you get:
    Simplify and you get a variable separable differential equation:
    Integrating and upon further simplification we get:
    y=c(1-x)/x +2 where c is an arbitrary constant.
    differentiate both side and plug in 1 in place of x and 2018 in place of y, we get
    Finally put everything, our function is:
    f(x)=-2018(1-x)/x +2
    Therefore f(2018)=2019

    • Elias Caeiro
      Elias Caeiro 10 months ago

      When you differentiate whit respect to x, you don't know if f is differentiable so you cannot do that for all x and integrate. You can only do this for x=1, or you can prove that f is differentiable.

  • Pedro Cardoso
    Pedro Cardoso 10 months ago

    Apply g(x)=xf(x) to get to
    Then, apply h(x)=g(x)-2 to get to
    Which means h(x)=m(x-1), where m is any real constant, and thus
    f(x)=(m(x-1)+2)/x, by reversing the substitutions and
    f'(x)=(m-2)/x^2, by differentiating.
    Since f'(1)=2018, we have m-2=2018 and m=2020
    Then f(2018) is (2020(2017)+2)/2018, which equals 2019.

  • Cobalt314
    Cobalt314 10 months ago

    The answer is 2019.
    First, setting y = 1, we have x(f(x) - f(1)) + 2x = xf(x) - f(1) + 2 -> x(f(1) - 2) = f(1) -2. As this must hold for all nonzero x, we require that f(1) = 2.
    Next, if we fix y and differentiate with respect to x (implicity), we get y(f(x) - f(y)) + xyf'(x) + 2 = f(x) + xf'(x), or f'(x) = (f(x) + yf(y) - yf(x) - 2)/(x(y-1)). Plugging in x = 1, we have f'(1) = (f(1) + yf(y) - yf(1) - 2)/(y-1) = y(f(y) - 2)/(y-1), and as it holds for all y, we solve it to get f(y) = (2020y - 2018)/y. Note that y = 1 is a removable discontinuity, as 1 - 1 = f(1) - 2. Thus, having the formula for f(y), we plug in 2018 to get f(2018) = (2020*2018 - 2018)/2018 = 2019.

  • Elias Caeiro
    Elias Caeiro 10 months ago

    My first solution to a weekly math challenge : ) :
    Let g(x) = f(x)/x. Our relation becomes yg(x) + g(y) + 2x = xg(y) + g(x) + 2y for all non-zero x, y. Let k = g(2)-2 g(2) = k+2.
    By plugging in y = 2, we get : 2g(x) + (k-2) + 4 = x(k-2) + g(x) + 2y g(x) = kx + 2 - k for all non-zero x.
    We have g'(x) = (f'(x)*x - f(x))/x^2 so g'(1) = f'(1)-f(1) = f'(1) - 1*g(1) = 2018 - (k*1 + 2 -k) = 2016. However, we also have g'(x) = k so g'(1) = k. Therefore k = 2016 ==> g(x) = 2016x - 2014 so f(x) = x(2016x - 2014).
    So f(2018) = 2018(2016*2018 - 2014) and we get f(2018) = 8205740932 using a calculator.

    • Elias Caeiro
      Elias Caeiro 10 months ago

      Sorry if I reply to my comment, I had a few calculations mistakes/typos… Here's the last part corrected :
      I meant of course g(x) = x*f(x) f(x) = g(x)/x.
      We have g'(x) = xf'(x) + f(x) so g'(1) = f'(1) + f(1) = f'(1) + g(1) = 2020. However, we also have g'(x) = k so g'(1) = k. Therefore k = 2020==> g(x) = 2020x - 2018.
      We get f(x) = (2020x-2018)/x = 2020 - 2018/x so f(2018) = 2019.

  • Minh Cong Nguyen
    Minh Cong Nguyen 10 months ago

    The answer is 2019
    Let f(-1)= c = constant, substituting y = -1 we have f(x)=(2x+2+xc-c)/(2x) then f'(x)= (c-2)/(2x^2)
    f'(1)=(c-2)/2=2018 which means that c=4038. Finally we have f(2018)=2019.

  • Minh Cong Nguyen
    Minh Cong Nguyen 10 months ago +1

    The answer is 2019

  • JHawk24
    JHawk24 10 months ago +1