Results 1 to 4 of 4

Thread: Find all functions

  1. #1
    Super Member dhiab's Avatar
    Joined
    May 2009
    From
    ALGERIA
    Posts
    582
    Thanks
    3

    Find all functions

    Find all functions f : IR ---> IR : ∀ (x,y) ∈ IRē :
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jan 2009
    Posts
    715

    If we replace $\displaystyle f(t) ~~$ by $\displaystyle \frac{1}{g(t)}$

    Then it becomes

    $\displaystyle \frac{1}{g(\frac{x+y}{2})}[ g(y) + g(x) ] = 2$

    $\displaystyle g(x) + g(y) = 2 g( \frac{x+y}{2})$

    It implies that $\displaystyle g(t) = t ~~$ so $\displaystyle f(t) = \frac{1}{t}$

    Is there another function satisifying the requirement ?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2009
    Posts
    91
    How about $\displaystyle f(t)=\frac1{at+b}$ for constants $\displaystyle a$, $\displaystyle b$ not both zero?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Mar 2009
    Posts
    91
    The equation $\displaystyle g\left(\frac{x+y}2\right)=\frac{g(x)+g(y)}2$ that occurs in simplependulum's reply is known as Jensen's equation.

    To solve it, put $\displaystyle y=0$ and then $\displaystyle g\left(\frac x2\right)=\frac{g(x)+b}2$ where $\displaystyle b=g(0)$.

    Put $\displaystyle x+y$ in this latter equation: $\displaystyle g\left(\frac{x+y}2\right)=\frac{g(x+y)+b}2$.

    And now we see that $\displaystyle g(x+y)+b=g(x)+g(y)$.

    Finally, put $\displaystyle h(x)=g(x)-b$. Then $\displaystyle h(x+y)=h(x)+h(y)$. This is the well-known Cauchy's equation.

    The most general continuous solution of Cauchy's equation is $\displaystyle h(x)=ax$ where $\displaystyle a$ is constant. Therefore $\displaystyle g(x)=ax+b$ is the most general continuous solution of Jensen's equation.

    However, it must be said that there exist solutions of Cauchy's equation on $\displaystyle \mathbb R$ which are not continuous, and they turn out to be extremely pathological. For example, if $\displaystyle h$ is one of these solutions and $\displaystyle I=(a,b)$ is any open interval then $\displaystyle h(I)$ is dense in $\displaystyle \mathbb R$.

    So it makes sense in many cases to ask for conditions on the solution, such as continuity, or boundedness on a finite interval, or monotonicity, anything which would avoid these weird functions.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. find all functions
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Jun 5th 2010, 06:30 AM
  2. Find all The Functions !
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: Feb 5th 2010, 03:50 PM
  3. Find all functions !
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Feb 1st 2010, 10:29 PM
  4. Find all functions
    Posted in the Algebra Forum
    Replies: 0
    Last Post: Aug 15th 2008, 12:53 AM
  5. Find Functions f and g
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: Jan 21st 2007, 04:45 AM

Search Tags


/mathhelpforum @mathhelpforum