Subtracting Functions

Subtracting functions subtracts their outputs. Take $p\left(x\right) = {x}^{2}+4x$ and $q\left(x\right) = 3x-5$. Using both rules turns $p\left(x\right)-q\left(x\right)$ into ${x}^{2}+4x-\left(3x-5\right)$, with q's whole rule in brackets.

The minus belongs to all of q, not only its first term. Taking away $3x-5$ means taking away 3x and taking away -5, which adds 5. So the line becomes ${x}^{2}+4x-3x+5$, and collecting gives ${x}^{2}+x+5$.

The usual slip is dropping the brackets and writing ${x}^{2}+4x-3x-5$. Check with x = 1: p(1) is 5 and q(1) is -2, so the difference is 7. The right rule gives 7; the slip gives -3.