Number of roots of the equation $f(xf(x)) = \sqrt{9 - x^2f^2(x)}$ 
Given the following graph of a $5$th degree polynomial $y = f(x)$:



Find the number of solutions of the equation: $$f(xf(x)) = \sqrt{9 - x^2f^2(x)}$$


*

*My attempt was to figure out what $f(x)$ is, by noticing the roots of its derivative based on the graph to know $f'(x)$, then find $\int f'(x)$, but the function I found did not match the original graph. (Even if I founded, I think it would be very messy to plug it in the equation)


*One other idea is that let $t = xf(x)$, then the equation become:
$$f(t) = \sqrt{9-t^2}$$
But I can't process from here.
Is my idea correct? If yes, then how do I continue? Or is there a better way to solve this?
 A: 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.
A: $f(x)=a_5x^5+\cdots+a_0\Rightarrow xf(x)=a_5x^6+\cdots+a_0x$ has degree $6$ so $$F(x)=f(xf(x))=a_5(a_5x^6+\cdots+a_0x)^5+\cdots+a_1(a_5x^6+\cdots+a_0x)+a_0$$ has degree $30$. We have $$(F(x))^2=9-(xf(x))^2$$ from which the resulting polynomial has degree $60$.
Thus there are $\color{red}{60}$ roots, not all distinct maybe because of possible multiplicity.
A: The number of roots seems to be very dependent on the $f(x) = a_0+a_1x+a_2x^2+a_3x³+a_4x^4+a_5x^5$ coefficients. Follows a MATHEMATICA script showing the results for an adjusted polynomial following the given figure.
f[x_] := a0 + a1 x + a2 x^2 + a3 x^3 + a4 x^4 + a5 x^5
equs = {f[0] == 2, f[3] == 0, f[-1] == 1, f[1] == 3, f[6] == 4, (D[f[x], x] /. {x -> 3}) == 0}
sol = Solve[equs, {a0, a1, a2, a3, a4, a5}][[1]]
f0 = f[x] /. sol
Plot[f0, {x, -2, 8}, AspectRatio -> 1, PlotRange -> {{-2, 7}, {-2, 8}}, PlotStyle -> {Thick, Blue}]


According to the required question the polynomial found is
$$
f(x) = -\frac{1219 x^5}{30240}+\frac{477 x^4}{1120}-\frac{32747 x^3}{30240}-\frac{477 x^2}{1120}+\frac{1189 x}{560}+2
$$
Now following
F[x_] := 2 + (1189 x)/560 - (477 x^2)/1120 - (32747 x^3)/30240 + (477 x^4)/1120 - (1219 x^5)/30240
error = F[x F[x]] - Sqrt[9 - x^2 F[x]^2]
Plot[error, {x, -2, 4}, PlotRange -> {{-1.5, 4}, {-3, 1}}, PlotPoints -> 50, PlotStyle -> {Thick, Blue}]


so we can count $10$ real roots located at
$$
\{-1.09006, 0.242585, 0.472128,\cdots,3.43159, 3.62582 \}
$$
