Solve Functional Equation on $\mathbb{Z}$

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In summary, a functional equation is an equation where the unknown variable is a function. It differs from a regular equation in that it involves finding a function as the solution. Solving a functional equation on the set of integers has significance because it allows for a function to be found that satisfies the given conditions for all integer values. Common methods for solving functional equations on Z include substitution, induction, and considering special cases or patterns. Additionally, there are many real-life applications of solving functional equations on Z, such as in economics, physics, and computer science.
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Let $ \mathbb{Z} $ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z} $ such that, for all integers $a$ and $b$, $f(2a)+2f(b)=f(f(a+b))$.
 
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Substituting $a=0$, $b=n+1$ gives $f(f(n+1))=f(0)+2f(n+1)$.

Substituting $a=1$, $b=n$ gives $f(f(n+1))=f(2)+2f(n)$.

In particular, $f(0)+2f(n+1)=f(2)+2f(n)$ and so $f(n+1)-f(n)=\dfrac{1}{2}\left(f(2)-f(0)\right)$.

Thus, $f(n+1)-f(n)$ must be constant. Since $f$ is defined only on $\mathbb{Z}$, this tells us that $f$ must be a linear function. Write $f(n)=Mn+K$ for arbitrary constants $M$ and $K$, and we need to only determine which choices of $M$ and $K$ work.

Now, $f(2a)+2f(b)=f(f(a+b))$ becomes $2Ma+K+2(Mb+K)=M(M(a+b)+K)+K$ which we may rearrange to form

$(M-2)(M(a+b)+K)=0$

Thus, either $M=2$ or $M(a+b)+K=0$ for all values of $a+b$. In particular, the only possible solutions are $f(n)=0$ and $f(n)=2n+K$ for any constant $K\in \mathbb{Z}$.
 

FAQ: Solve Functional Equation on $\mathbb{Z}$

What is a functional equation?

A functional equation is an equation in which the unknowns are functions rather than simple variables. It relates the values of a function at different points to each other, rather than just relating the values to a fixed constant.

What is the difference between a functional equation and a regular equation?

The main difference is that a functional equation involves unknown functions, while a regular equation involves unknown variables. In a functional equation, we are trying to find the function that satisfies the equation, while in a regular equation, we are trying to find the value of a variable that satisfies the equation.

What is the importance of solving functional equations on the set of integers (Z)?

Solving functional equations on the set of integers is important because it allows us to understand the behavior of functions on a discrete set of numbers. This is useful in various fields such as computer science, number theory, and combinatorics.

What are some common techniques for solving functional equations on the set of integers?

Some common techniques for solving functional equations on the set of integers include substitution, induction, and using properties of modular arithmetic. Other techniques such as graphing and trial and error can also be useful in certain cases.

Are there any real-life applications of solving functional equations on the set of integers?

Yes, there are various real-life applications of solving functional equations on the set of integers. For example, in computer science, functional equations can be used to analyze algorithms and data structures. In economics, functional equations can be used to model supply and demand relationships. In cryptography, functional equations can be used to design secure encryption algorithms.

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