How is f(-x) related to f(x) ?jd12345 said:Homework Statement
A function from R-->R is differentiable and follows f( (x+y)/3 ) =( 2 + f(x) + f(y) ) / 3
Derivative of f(x) at x=2 is 2
Find the range of f ( |x| )
Homework Equations
The Attempt at a Solution
Well the questions asks me the range of f( |x| ). But i don't even know f(x). I did find the value of f(0) which is 2. And i don't know what to do afterwards. I have no idea why the value of derivative is given.
If f(x) is odd, then it is true that f '(x) is even, but the converse is not necessarily true.jd12345 said:Ok (by using spoiler) f(0) = 2 + f(x) + f(-x) / 3
=> 6 = 2 + f(x) +f(-x)
=> 4 = f(x) +f(-x)
Thus 0 = f '(x) - f '(-x)
f '(x) = f '(-x)
f '(x) is even so f(x) is odd. This is not true.
But if f(x) is odd then f(x) + f (-x) = 0 but i found out earlier that it is equal to 4. A bit confused
SammyS said:Two more substitutions that might be helpful:
y = x (which you may have used to find f(0) )
and
y = 2x .
A functional equation is an equation where the unknown variable is a function, rather than a traditional numerical value. This means that the solution to the equation is a function that satisfies certain conditions or relationships.
Solving functional equations is important in science because many natural phenomena and relationships can be described using functional equations. By solving these equations, we gain a better understanding of the underlying mechanisms and can make predictions about future behavior.
There are several strategies commonly used to solve functional equations, including substitution, differentiation, and induction. It is also important to carefully analyze the given conditions and properties of the function in order to identify any patterns or relationships that can help in finding a solution.
No, not all functional equations can be solved analytically. Some equations may require numerical or graphical methods to approximate a solution, while others may not have a solution at all. In these cases, it is important to understand the limitations of the equation and interpret the results accordingly.
Functional equations can be applied in a variety of real-world situations, such as modeling population growth, predicting market trends, and understanding physical systems. By using functional equations, we can make accurate predictions and inform decision-making processes in various fields of study.