Number of roots of the equation $f(xf(x)) = \sqrt{9 - x^2f^2(x)}$
According to the given question, you need to find the solutions of $$f(xf(x)) = \sqrt{9 - x^2f^2(x)}$$
Substituting $x(f(x))$ by $t; t\in (-\infty , \infty)$, like you did, we obtain
$$f(t) = \sqrt{9-t^2} \\ \Rightarrow f(x) = \sqrt{9-x^2}$$
ie., We need to find the number of points where the graph of $f(x)$ and $ \sqrt{9-x^2} =g(x) \space \text{(say)}$ intersect.
Analysing $g(x)$, we can say that $$ x^2 + \left(g(x)\right)^2 =9 ; g(x) \geq 0$$ ie., $g(x)$ traces a semicircular region as shown below
Superimposing the graphs of $f(x)$ and $g(x)$ we obtain,
(Zoomed in view)
As we can notice four points of intersection, there are four solutions to the given expression $f(t) = \sqrt{9 - t^2}$
Now, the values of $t$ that satisfy are
- $t$ approximately $-1.5$ or something close to that, which the product $xf(x)$ can give in 2 different ways. First when $x<0$, second when $f(x)<0$.
- $t = 3$ which can be obtained in 4 ways, once for $x=1, f(x)=3$ then, for $x=3, f(x)=1$ and twice for $x<1$.
- once more check it for $t$ between $0$ and $1$
- and lastly for $t$ between $1$ and $2$.
Edit #1: The last two calculations would get messier the more we try to analyse it raw-handedly. Holding on to the suggestion by @windows prime in the comments, we can draw an approximate graph of $t= x f(x) =t_{\small 0}$ having $t_{\small 0}$ as a solution of $t$
These graphs would be rectangular hyperbolas if you observe them clearly with $f(x)$ at $y-axis$.
For both, $t_1$ whose value lies between $0$ and $1$ and $t_2$ whose value lies between $1$ and $2$, the graphs will have 4 intersections each with that of $y= f(x)$.
So, we have $2+4+4+4 = 14$ solutions in total.
Edit #2: The intersection of hyperbolic graphs and $f(x)$ would look approximately like the one below:
$\color{#Ff3537}{xy \approx -1.5}\\ \color{#4466ff}{xy = 3}\\ \color{#008800}{xy = t_1;\space t_1\in (0,1)}\\ \color{#8800ff}{xy = t_2;\space t_2\in(1,2)}\\$
Here, we have 14 intersections pointing towards 14 distinct solution.