NapkinCalc

Precalculus A — Functions & Transformations

Function notation

ELI5: f(x) = x² − 3x names a machine; f(4) runs it (plug 4 in everywhere x appears). Once defined, every cell below can call it — no retyping the formula.

f(x)=x2−3⋅xf\left(x\right) = x^{2} - 3 \cdot x define the machine
ya=f(4)y_{a} = \mathrm{f}\left(4\right) = 44 run it: 16 − 12 = 4
yb=f(−1)y_{b} = \mathrm{f}\left(-1\right) = 44 negative inputs are fine: 1 + 3 = 4

Real-world hook: a function is any fixed rule — price as a function of weight, tax as a function of income, braking distance as a function of speed.

Try it yourself: define p(x) = x² + 1, then we run it: the check confirms p(3) = 10 and p(0) = 1.

p(x)=x2+1p\left(x\right) = x^{2} + 1 ✏️ Your turn: define the machine p(x) = x² + 1. Green when running it gives p(3) = 10 and p(0) = 1.
✓ pass abs(p(3)−10)<1e−9andabs(p(0)−1)<1e−9abs(p(3) - 10) < 1e-9 and abs(p(0) - 1) < 1e-9 green when your machine produces 10 and 1