Coupon First or Discount First?

A store offers 20% off and you also have a $10 coupon. Taking 20% off is one function, subtracting$ 10 is another, and checking out applies one after the other: a composition. Swapping the order changes what you pay.

How it works

Composition of functions, and why order matters

In one sentence

The composition of $f$ and $g$ is $(f \circ g)(x) = f(g(x))$ , defined where $g(x)$ lies in the domain of $f$ . In general $f \circ g \ne g \circ f$ ; for $d(x) = (1 - p)x$ and $c(x) = x - k$ , $(d \circ c)(x) - (c \circ d)(x) = pk$ .

General formula

{\htmlData{sym=0}{d(x)}} = (1 - {\htmlData{sym=3}{p}})\,{\htmlData{sym=2}{x}}, \quad {\htmlData{sym=1}{c(x)}} = {\htmlData{sym=2}{x}} - {\htmlData{sym=4}{k}}, \qquad d({\htmlData{sym=1}{c(x)}}) = (1 - {\htmlData{sym=3}{p}})({\htmlData{sym=2}{x}} - {\htmlData{sym=4}{k}}), \quad c({\htmlData{sym=0}{d(x)}}) = (1 - {\htmlData{sym=3}{p}})\,{\htmlData{sym=2}{x}} - {\htmlData{sym=4}{k}}

Composition is a pipeline: the output of the first step is the input to the second. Change the order of the steps and the result can change.

What the symbols mean

$d(x)$
Price after the discount
What a price x becomes after the percentage off.
$c(x)$
Price after the coupon
What a price x becomes after subtracting the coupon.
$x$
Price
The price going into each step.
$p$
Discount rate
The fraction taken off: 0.2 means 20% off.
$k$
Coupon value
The dollar amount the coupon removes.

Each deal is a function of the price: the discount $d(x) = (1 - p)x$ and the coupon $c(x) = x - k$ .
Coupon first, then discount, is the composition $d(c(x)) = (1 - p)(x - k)$ : the output of $c$ becomes the input of $d$ .
The other order is $c(d(x)) = (1 - p)x - k$ . Expanding the first gives $(1 - p)x - (1 - p)k$ , so the two always differ by exactly $pk$ .
Composition isn't commutative: d∘c and c∘d are different functions. Here discounting first is always cheaper, because the whole coupon then comes off the already-reduced price.

Learn the math

Combining FunctionsAlgebra · Lesson 25, Graphing and Functions