How a Ride-Share Fare Is Computed

A ride costs a base fee plus a charge per mile, but never less than a minimum fare, and long trips get a lower per-mile rate after a set distance. Each rule applies on its own interval of distances: a piecewise function.

How it works

Piecewise-defined functions

In one sentence

A piecewise-defined function assigns a formula to each interval of its domain; to evaluate it, find the interval containing the input and use that formula. It's continuous at a boundary when the neighboring formulas give the same value there.

General formula

{\htmlData{sym=0}{F(d)}} = \begin{cases} \max\left({\htmlData{sym=1}{F_{\min}}},\; {\htmlData{sym=2}{b}} + {\htmlData{sym=3}{m}}{\htmlData{sym=4}{d}}\right) & 0 \le {\htmlData{sym=4}{d}} \le {\htmlData{sym=5}{D}} \\ {\htmlData{sym=2}{b}} + {\htmlData{sym=3}{m}}{\htmlData{sym=5}{D}} + {\htmlData{sym=6}{m_2}}({\htmlData{sym=4}{d}} - {\htmlData{sym=5}{D}}) & {\htmlData{sym=4}{d}} > {\htmlData{sym=5}{D}} \end{cases}

Each line of the price list is a separate formula with its own range of distances. The graph is those rules stitched together.

What the symbols mean

$F(d)$
Fare
What a trip of d miles costs.
$F_{\min}$
Minimum fare
The least any ride can cost.
$b$
Base fee
Charged at pickup, before any miles.
$m$
Per-mile rate
Dollars per mile on the first D miles.
$d$
Trip length
Miles traveled.
$D$
Long-trip distance
Where the cheaper per-mile rate kicks in.
$m_2$
Long-trip rate
Dollars per mile beyond D miles.

A piecewise function uses different formulas on different intervals of its input; here the input is the trip distance $d$ .
Short trips pay the minimum fare $F_{\min}$ until the base fee plus the per-mile charge, $b + md$ , passes it. That's the flat start of the graph.
After $D$ miles the per-mile rate drops to $m_2$ , so the graph continues from where it was with a gentler slope.
The pieces meet at their boundaries, so the fare never jumps: the function is continuous.

Learn the math

The Definition of a FunctionAlgebra · Lesson 24, Graphing and Functions Piecewise Functions and the Basic GraphsAlgebra · Lesson 27, Common Graphs