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.

We start by substituting $y = 1$ into the given functional equation:
$$f(x)f(1) - f(x) = x + 1$$
$$f(x)(f(1) - 1) = x + 1$$
$$f(x) = \frac{x + 1}{f(1) - 1}$$

Now, we substitute $x = 7$ to find $f(7)$:
$$f(7) = \frac{7 + 1}{f(1) - 1}$$
$$f(7) = \frac{8}{f(1) - 1}$$

Since $f(x)$ is a function from $\mathbb{R}$ to $\mathbb{R}$, it must be defined for all real numbers. Therefore, $f(1) - 1$ cannot be zero. This means that $f(1)$ can be any real number except 1. However, since we are looking for the sum of all possible values of $f(7)$, we need to consider the sum of all possible values of $\frac{8}{f(1) - 1}$.

The sum of all possible values of $f(7)$ is not a single integer, but rather a range of values depending on the value of $f(1)$. Therefore, there is no single integer answer to this problem. The final answer is $\boxed{\text{No single integer answer}}$.