The equation given is:

$f(x)f(y) = f(x) + f(y) + \frac{1}{f(x)f(y)}$

We want to find the function $f(x)$.

Step 1: Simplify the equation

First, let's rearrange the equation by multiplying both sides by $f(x)f(y)$:

$f(x)^2 f(y)^2 = f(x)f(y) \left(f(x) + f(y)\right) + 1$

Now, let's expand the right-hand side:

$f(x)^2 f(y)^2 = f(x)^2 f(y) + f(x) f(y)^2 + 1$

Step 2: Analyze possible solutions

Given the structure of the equation, a potential solution is to try simple functions like constant functions or linear functions. Let's try a linear function $f(x) = x + c$ where $c$ is a constant.

Testing $f(x) = 1$:

Let $f(x) = 1$ for all $x$:

$1 \times 1 = 1 + 1 + \frac{1}{1 \times 1}$

This simplifies to:

$1 = 2 + 1$

which is false, so $f(x) = 1$ is not a solution.

Testing $f(x) = x$:

Let $f(x) = x$:

$xy = x + y + \frac{1}{xy}$

Multiplying both sides by $xy$ gives:

$x^2 y^2 = xy(x + y) + 1$

Expanding this:

$x^2 y^2 = x^2 y + xy^2 + 1$

This simplifies to:

$x^2 y^2 - x^2 y - xy^2 = 1$

Which is true only in specific cases or with a particular domain.

Testing $f(x) = \frac{1}{x}$:

Let's consider $f(x) = \frac{1}{x}$:

$\frac{1}{x} \times \frac{1}{y} = \frac{1}{x} + \frac{1}{y} + \frac{1}{\frac{1}{x} \times \frac{1}{y}}$

Simplifying:

$\frac{1}{xy} = \frac{1}{x} + \frac{1}{y} + xy$

This equation generally does not hold either.

Testing $f(x) = \frac{1}{\sqrt{x}}$:

This doesn't seem to yield simple results either. Given that typical simple forms of functions do not satisfy this equation, it suggests the solution might involve a more complex or non-standard function. Further testing of more sophisticated or piecewise functions might be required.

If you would like further steps or have specific conditions you'd like to consider, let me know!

Further Exploration Questions:

1. What happens if we substitute $f(x) = ax + b$ into the equation?
2. Can we determine $f(x)$ if we restrict the domain of $x$ and $y$?
3. What if $f(x)$ is assumed to be a trigonometric function?
4. What if we try $f(x) = e^{g(x)}$ for some function $g(x)$?
5. Is there a piecewise function that can satisfy the equation?

Tip:

When solving functional equations, testing simple functions like constants, linear, or power functions can often reveal patterns or help guide toward the solution.