f(y+f(x)) = x + f(y+2024) given f(0)=2024 Find f(x)

We are given the functional equation:

\[
f(y + f(x)) = x + f(y + 2024)
\]

with the condition \( f(0) = 2024 \). We need to find the function \( f(x) \).

### Step 1: Let \( y = 0 \)

Substituting \( y = 0 \) into the equation, we get:

\[
f(0 + f(x)) = x + f(0 + 2024)
\]
\[
f(f(x)) = x + f(2024)
\]

Let’s call \( f(2024) = c \), so the equation becomes:

\[
f(f(x)) = x + c
\]
This is an important relation that we'll use later.

### Step 2: Let \( x = 0 \)

Substituting \( x = 0 \) into the original equation, we get:

\[
f(y + f(0)) = 0 + f(y + 2024)
\]
Since \( f(0) = 2024 \), this becomes:

\[
f(y + 2024) = f(y + 2024)
\]

This equation is trivially true and doesn’t provide any new information.

### Step 3: Analyzing the structure of the functional equation

We already have:

\[
f(f(x)) = x + c
\]
This suggests that \( f(x) \) is a linear function. Let’s assume \( f(x) = x + 2024 \) and check if it satisfies the original equation.

### Step 4: Check if \( f(x) = x + 2024 \) is a solution

Substitute \( f(x) = x + 2024 \) into the original equation:

The left-hand side of the functional equation is:

\[
f(y + f(x)) = f(y + (x + 2024)) = (y + x + 2024) + 2024 = y + x + 4048
\]

The right-hand side is:

\[
x + f(y + 2024) = x + (y + 2024 + 2024) = x + y + 4048
\]

Both sides are equal, so \( f(x) = x + 2024 \) is indeed a solution.

### Final Answer:
The function is \( \boxed{f(x) = x + 2024} \).

Would you like a detailed explanation or have any questions?

Here are some related questions to deepen understanding:

1. How do we identify linear functional equations?
2. Why did we assume \( f(x) \) is linear in Step 3?
3. Can we try other forms for \( f(x) \) besides linear?
4. How would this problem change if \( f(0) \) had a different value?
5. How can we apply this method to more complex functional equations?

**Tip**: Always start by trying simple values like \( x = 0 \) or \( y = 0 \) to simplify functional equations.

f(y + f(x)) = x + f(y + 2024) given f(0)=2024. Find f(x).

Discover how to solve the functional equation f(y + f(x)) = x + f(y + 2024) with the condition f(0) = 2024. This step-by-step solution explores key mathematical concepts like functional equations and linear functions. Suitable for students in Grades 10-12.

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