Piecewise functions as models

Piecewise functions are handy for writing real-world models with different rules on different intervals: tiered tariffs, deformation of a material before and after the elastic limit, the cost of a utility with a free allowance. Solving an equation $f(x)=k$ or an inequality $f(x)\lessgtr k$ is done piece by piece, checking at the end the compatibility with the interval of each piece.

Example — Tiered tariff

A telephone company charges: $0{,}10$ €/min for the first $100$ minutes of the month, $0{,}05$ €/min beyond that. Total cost as a function of the minutes $m$: $C(m)=\begin{cases} 0{,}10\,m & 0\le m\le 100, \\ 10 + 0{,}05(m-100) & m>100.\end{cases}$ For how many minutes does one pay $C=15$ euros? On $0\le m\le 100$: $0{,}10\,m=15 \iff m=150$, outside the interval. On $m>100$: $10+0{,}05(m-100)=15 \iff m-100=100 \iff m=200$. Solution: $\boxed{m=200}$ minutes.

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Topics: Functions and properties
Concepts: Piecewise function
Skills: Modelling · Solving equations
Exercise types: Modelling problem