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In general, function composition is not commutative, which means that $(f \circ g)(x) \neq (g \circ f)(x)$.

If $f: A \rightarrow B$ and $g: B \rightarrow C$ are both bijective, then the composition $(f \circ g): A \rightarrow C$ is also bijective.

The range of the composition $(f \circ g)(x)$ is determined by the values that $f(g(x))$ can take, which depends on the range of $g$ and the function $f$.

If $f$ and $g$ are inverse functions, then $f \circ g = id_x$ and $g \circ f = id_y$, where $id_x$ and $id_y$ are identity functions on respective sets.

The identity function, denoted as $id_x$, is a function that always returns its input: $id_x(x) = x$.

The composition of two functions, where you apply one function to the result of another function.

The domain of the composition $(f \circ g)(x)$ is the set of all values $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.