Saturday, April 6, 2024

CHATGPT SOLUTION

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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
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 .

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