Composing Rules

A composition can be written as one new rule. Take $p\left(x\right) = x+1$ and $q\left(x\right) = {x}^{2}+4$. In $q\left(p\left(x\right)\right)$, the output of p is the expression $x+1$, and it goes into q in place of x, whole and in brackets: $q\left(p\left(x\right)\right) = {\left(x+1\right)}^{2}+4$.

Then multiply out and collect: ${\left(x+1\right)}^{2}$ is ${x}^{2}+2x+1$, so the rule is ${x}^{2}+2x+5$.

Check with one number. At x = 2, p(2) is 3 and q(3) is 13. The new rule gives $4+4+5$, which is 13 too. One rule now does the work of two.