A Function of Itself

A function can be composed with itself. $h\left(h\left(x\right)\right)$ means: apply h, then apply h again to the output. Take $h\left(x\right) = 3x-1$ and work out $h\left(h\left(1\right)\right)$. First h(1) is 2. Then h(2) is 5.

Written out, the inner call comes first and gives $h\left(3\left(1\right)-1\right)$, the outer one $3\left(3\left(1\right)-1\right)-1$, and the arithmetic finishes it: $3\left(2\right)-1$, then 5.

Do not confuse this with squaring or doubling. ${\left(h\left(1\right)\right)}^{2}$ is 4 and $2h\left(1\right)$ is 4, but $h\left(h\left(1\right)\right)$ is 5. Applying a rule twice means running the second pass on the first pass's output, the way a loop feeds each result back in.