Combining Functions: Sums, Quotients and Composition

Building a model from pieces

A school club sells T-shirts for $12$ dollars each. Printing costs a $500$-dollar setup fee plus $4$ dollars a shirt.

$$R(x) = 12x \qquad C(x) = 500 + 4x$$

Profit is what revenue leaves after cost, so the profit function is the difference of the two:

$$P(x) = R(x) - C(x) = 12x - (500 + 4x) = 8x - 500$$

The club breaks even when $P(x) = 0$, at $x = 62.5$, so it needs to sell $63$ shirts to come out ahead.

Functions combine the way numbers do, one input at a time:

Combination	Rule
sum	$(f + g)(x) = f(x) + g(x)$
difference	$(f - g)(x) = f(x) - g(x)$
product	$(f \cdot g)(x) = f(x) \cdot g(x)$
quotient	$\left(\tfrac{f}{g}\right)(x) = \tfrac{f(x)}{g(x)}$, where $g(x) \ne 0$

A cooling cup of coffee

Coffee poured at $180°$F sits in a $70°$F room. It cools fast at first and then more slowly, and it never drops below the temperature of the room.

The model is a sum of two familiar functions:

$$T(t) = 70 + 110(0.9)^t$$

The $70$ is a constant function, the room temperature. The $110(0.9)^t$ is a decaying exponential: the coffee starts $110$ degrees above the room, and each minute that excess shrinks by $10\%$.

• T(t), °F
• room, 70°F

$t$ (minutes)	$0$	$5$	$10$	$20$	$30$
$T(t)$ (°F)	$180$	$135.0$	$108.4$	$83.4$	$74.7$

Why adding functions adds their graphs

$(f + g)(x)$ is defined one input at a time: at each $x$, take the height of one graph and the height of the other and add them. So the graph of the sum is the two graphs stacked, point by point.

That is exactly what the coffee graph shows. The decaying exponential $110(0.9)^t$ on its own heads toward $0$. Adding the constant $70$ lifts every point of it by $70$, so the combined curve heads toward $70$ instead.

Each part of the model has one job, and it can be read off:

Part	Job	Shows up as
$70$	the room	the level the curve approaches
$110$	the starting excess	$T(0) - 70$
$0.9$	the cooling rate	$10\%$ of the excess lost per minute

Building a model this way makes every number in it mean something. A single formula found by curve-fitting might match the data as well, but it would not say why the coffee stops at $70$.

A quotient: average cost

The club’s cost per shirt is total cost divided by the number of shirts:

$$A(x) = \frac{C(x)}{x} = \frac{500 + 4x}{x} = \frac{500}{x} + 4$$

Shirts	$10$	$50$	$100$	$500$	$1000$
Average cost	$54.00$	$14.00$	$9.00$	$5.00$	$4.50$

The setup fee gets shared among more shirts as $x$ grows, so the $\tfrac{500}{x}$ part shrinks toward $0$ and the average cost falls toward the $4$ dollars each shirt costs to print. The quotient excludes $x = 0$, where no shirts means no average.

Composition: one output feeds the next

Sometimes one quantity depends on a second, which depends on a third.

A weather balloon rises $500$ feet a minute, so its height after $t$ minutes is $h(t) = 500t$. The air cools with height; near the ground a good model is $T(h) = 59 - 0.0036h$ degrees Fahrenheit, about $3.6°$ per $1{,}000$ feet.

To get temperature as a function of time, feed the height into the temperature function:

$$T(h(t)) = 59 - 0.0036(500t) = 59 - 1.8t$$

This is a composition, written $(T \circ h)(t) = T(h(t))$. Apply $h$ first, then $T$ to the result. After $10$ minutes the balloon is at $5{,}000$ feet and the air around it is $59 - 18 = 41°$F.

Worked examples

Common mistakes

Practice problems