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CHATGPT SOLUTION

You

x and y are Real. f(x) and f(y) are Real. Find f(x) and f(y) if f(f(x+y) = f(x) + f(y)

ChatGPT

To find $�(�)$ and $�(�)$, we need to solve the functional equation $�(�(�+�))=�(�)+�(�)$.

Let's denote $�(�)$ as $�$ for simplicity:

$�(�(�+�))=�(�)+�(�)$

Now, let's consider some particular values of $�$ and $�$ to simplify the equation:

1. Let $�=�=0$: $�(�(0))=�(0)+�(0)$ $�(�(0))=2�(0)$

2. Let $�=0$: $�(�(�))=�(�)+�(0)$

3. Let $�=0$: $�(�(�))=�(0)+�(�)$

Now, let's compare the equations from 2 and 3:

$�(�)+�(0)=�(0)+�(�)$

This implies that $�(0)$ is a constant, let's denote it as $�$. Therefore:

$�(�)+�=�+�(�)$ $�(�)=�$

So, $�(�)=�$ for all real values of $�$.

Therefore, $�(�)$ and $�(�)$ are both constant functions, and the value of $�(�)$ and $�(�)$ is the same for any real numbers $�$ and $�$.