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# Composition is not commutative

Contrary to the properties of sums and products, order matters in composition. In general, $$f\circ g\ne g\circ f.$$ We say that composition is not commutative.

Setting $f(x) = x^2$ and $g(x) = x+1$, we see that, for $x\ne0$, $$f\circ g(x) = (x+1)^2\ne x^2+1 = g\circ f(x).$$

However, for the inverse function, we have $$f\circ f^{-1}(y) =y,\qquad f^{-1}\circ f(x) =x$$ This comes from $y=f(x)\Longleftrightarrow x=f^{-1}(y)$.

The composition of monotonic functions is monotonic. If both functions have same monotonicity (both increasing or decreasing), the composite function is increasing; otherwise it is decreasing.

To see this, assume that both $f$ and $g$ are decreasing. Then, for $x\le y$, we have $g(x)\ge g(y)$, hence $f\left(g\left(x\right)\right)\le f\left(g\left(x\right)\right)$. This means that means that $f\circ g$ is increasing.