We are given that $f$ is a linear function and satisfies the functional equation:

$f(x + f(x)) = x + f(x)$

We are tasked with determining $f(5)$.

Step 1: Analyze the functional equation

Let’s first rewrite the given equation for clarity:

$f(x + f(x)) = x + f(x)$

Since this holds for any $x$, we can consider special cases or deduce properties of the function $f$. Let’s assume $f(x)$ is a linear function, which means we can write it in the form:

$f(x) = ax + b$

where $a$ and $b$ are constants. We will substitute this expression into the functional equation and solve for $a$ and $b$.

Step 2: Substitution into the functional equation

Substituting $f(x) = ax + b$ into the equation $f(x + f(x)) = x + f(x)$:

$f(x + f(x)) = f(x + ax + b) = a(x + ax + b) + b = ax + a^2x + ab + b$

On the right-hand side, we have:

$x + f(x) = x + ax + b$

Now equate both sides:

$ax + a^2x + ab + b = x + ax + b$

Step 3: Simplifying

We can simplify this equation by collecting like terms:

$a^2x + (ab + b) = x$

Now, comparing the coefficients of $x$ and the constant terms:

• Coefficient of $x$: $a^2 = 1$, so $a = \pm 1$.
• Constant term: $ab + b = 0$, which simplifies to $b(a + 1) = 0$.

Step 4: Solve for $a$ and $b$

1. If $a = 1$, the equation $b(1 + 1) = 0$ gives $b = 0$.

   Thus, $f(x) = x$.

2. If $a = -1$, the equation $b(-1 + 1) = 0$ is automatically satisfied for any $b$. Therefore, $f(x) = -x + b$.

Step 5: Determine $f(5)$

We now have two possible forms for $f(x)$:

1. If $f(x) = x$, then $f(5) = 5$.
2. If $f(x) = -x + b$, we need additional information about $b$ to determine $f(5)$. Since the problem doesn’t provide more constraints, both forms are potential solutions, but $f(x) = x$ is the simplest and most natural solution under the assumption of linearity.

Conclusion

Thus, $f(5)$ can be equal to $5$.

Would you like a detailed exploration of any part of this solution? Here are some further questions you could explore:

1. What would happen if the function was not linear?
2. How does the assumption of linearity simplify the functional equation?
3. Can we derive any additional properties of the function based on this result?
4. What if we impose other conditions on the function $f$?
5. How would the solution change if $f$ were quadratic or of a higher degree?

Tip: Always verify assumptions like linearity when solving functional equations, as they can significantly simplify the process!