Adding Functions

Two functions can be added to make a new one. The sum $p\left(x\right)+q\left(x\right)$ means: give both functions the same input and add their outputs. Take $p\left(x\right) = 2x+5$ and $q\left(x\right) = {x}^{2}+3x$.

Using both rules gives $2x+5+{x}^{2}+3x$. Then collect like terms: 2x and 3x make 5x, so $p\left(x\right)+q\left(x\right) = {x}^{2}+5x+5$.

Collecting matters because the new rule should do in one step what the two rules did in two. Check with x = 1: p(1) is 7 and q(1) is 4, so the sum is 11, and the new rule gives $1+5+5$, which is 11 too.