functions - Functional equation on $mathbb{R}^+$

Let $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ be a function satisfying
$$f(x)f(yf(x))=f(x+y),\forall x,y\in\mathbb{R}^+$$

If $f(1)=\frac{1}{152}$, evaluate $f(4)$.

---

By inspection, we can see $f(x)=\frac{1}{151x+1}$ is a solution, from which we can easily get the answer. But how can we show that this is the only solution?

Here is my work:

Because $f$ is the reciprocal of a linear function, it would probably help to define $g(x)=\frac{1}{f(x)}$ (note that this is well defined as we are working in $\mathbb{R}^+$). Then the given equation becomes
$$g(x)g\left(\frac{y}{g(x)}\right)=g(x+y)$$

If we take $y=g(x)$, then this becomes
$$g(x)g(1)=g(x+g(x))\implies 152g(x)=g(x+g(x))$$

Not sure where to go from here. Any thoughts?

Answer

I'm going to proceed with your $g(x) = 1/f(x)$, starting with this
$$g(x)g\left(\frac{y}{g(x)}\right)=g(x+y)$$
but then re-setting in it $y = g(x)y_1$ (possible for any positive $y_1$) to get this basic equation
$$\tag1

g(x)g(y)=g(x+g(x)y)$$
(in which I renamed $y_1$ back to $y$). Applying this first with $x=1$ and then with some other $x = w > 1$, letting $B := g(w)$, gives
$$
g(1+152y) = 152g(y)\\
g(w+By) = Bg(y)
$$
Suppose $B < 152$. We can then pick a positive $y$ such that $1+152y = w + By$ so that the left-hand sides are equal, but the right-hand sides are not (since $g(y) > 0$) - contradiction. So $w > 1$ implies $g(w) \geq 152$.

Now consider $u = w + B(1+152y_0)$ and $v = 1+152(w+By_0)$ for some arbitrary $y_0 > 0$:
$$

g(u) = g(w + B + 152By_0) = Bg(1+152y_0) = 152Bg(y_0)\\
g(v) = g(1+ 152w +152By_0) = 152g(w+By_0) = 152Bg(y_0)
$$
Since the right-hand-sides are equal, $g(u) = g(v)$. This means that either $w+B = 1+152w$, meaning that
$$\tag2
g(w) = B = 1+151w
$$
as we wanted to prove, or, $g(u) = g(v)$ with $u \neq v$. We derive a contradiction in the latter case. WLOG, suppose $u < v$. Then setting in (1) $x = u, y = (v-u)/g(u) =: t$, we get
$$
g(u)g(t) = g(u + g(u)t) = g(v)\\

\therefore g(t) = 1
$$
Now set in (1) $x = t$, to get $g(t+y) = g(y)$ and since $g(t) = 1$, it follows by induction that $g(nt) = 1$ for any positive integer $n$. By choosing a sufficiently large $n$, we can make $nt > 1, g(nt) = 1$ which is a contradiction with the previous result that $w > 1 \Rightarrow g(w) \geq 152$. This leaves (2) as the unique solution.