Self composition

Just like we can compose two functions $f$ and $g$ to get $f g,$ we can also compose a function with itself. We denote this function $f^{2} = f f$ and this is different from squaring the original function.

Example

For example, if $f (x) = 2 x + 3,$ then $f^{2} (x) = f f (x) = f (2 x + 3) = 2 (2 x + 3) + 3 = 4 x + 9,$ which is not the same as $(f (x))^{2} = (2 x + 3)^{2} .$

In a similar fashion to the existence of $f g,$ we also have that the composite function $f^{2}$ exists if $R_{f} \subseteq D_{f} .$

A note about notation

We have just seen that $f^{2} (x)$ stands for $f f (x)$ and is different from the square of $f$ which we denote $(f (x))^{2} .$ In a similar fashion, we note that the previous section on inverse functions use the notation $f^{- 1} (x)$ to denote the “reverse” and is different from the reciprocal $(f (x))^{- 1} = \frac{1}{f (x)} .$