math-func-01

# A Multiplicative Functional Equation

Let $f:\mathbb{R}\to\mathbb{R}$ be a function satisfying
$$f(x)\,f(y) - f(xy) = x + y$$
for **all** real numbers $x$ and $y$.

Find the sum of all possible values of $f(7)$.

Give your final answer as a single integer.

The functional equation \( f(x)f(y) - f(xy) = x + y \) for all real numbers \( x \) and \( y \) can be solved by assuming \( f \) is a linear function. By substituting specific values and solving, we find that the only valid solution is \( f(x) = x + 1 \). Therefore, \( f(7) = 8 \).

The sum of all possible values of \( f(7) \) is:

\boxed{8}