Can I conclude that $x=f(x)$ from the assumption that $f(x)=f(f(x))$?

Just write it down from definitions. If $f$ is injective, then this means, by definition, that

$$\forall x_1, x_2\in D_f: f(x_1)=f(x_2)\implies x_1=x_2.$$

Now, set $x_1=x$ and $x_2=f(x)$. What does the above expression change into?