Given $f:[0,1]\rightarrow \mathbb{R}$ and $g:[0,1]\rightarrow [0,1]$, $g(x)=x^2$.
Which of the two equalities is true?
1)$f^2(x^2)=f^2(g(x))=(f^2\circ g)(x)$;
2)$f^2(x^2)=f(x^2)\cdot f(x^2)=f(g(x))\cdot f(g(x))=(f\circ g)(x)\cdot (f\circ g)(x)=(f\circ g)^2(x).$
Note: here $f^2$ is defined as the 2-fold product of $f$ and not composition.
The first equality is true, because $f^2(x)$ is usually defined to be $(f \circ f)(x) = f\big(f(x)\big)$, and $(f \circ g)(x)$ is defined as $f\big(g(x)\big)$. Because of these definitions, the second equality does not hold. However, if the notation $f^2$ or $\circ$ is differently defined, any one or possibly both equalities may be true.