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Saturday, April 6, 2024

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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:

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 $�$ .

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