Functions Require Context

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In this section we demonstrate that a relation requires context to be considered a function.

In the previous section we established that a relationship is a function if each input has exactly one output. This condition can be even trickier than it may initially seem though. The same equation can be a function in one setting, and not a function in another. This is because every relationship requires context before we can decide if it is a function or not. Consider the following example.

A Drink Vending MachineYou are thirsty and decide to get a drink from the vending machine nearby. After looking over your choices you see that the vending machine is setup like the following:

	#1	#2	#3	#4	#5

A	Pepsi	Pepsi	Pepsi	Pepsi	Pepsi
B	Fanta	Fanta	Sierra Mist	Sierra Mist	Sierra Mist
C	Gatorade Blue	Gatorade Blue	Gatorade Green	Gatorade Green	Gatorade Green
D	Coke	Coke	Coke	Coke	Coke
E	Sprite	Sprite	Sprite	Sunkist	Sunkist
F	Crush	Root beer	Cream Soda	Water	Water

If you punch in a letter and number combination you know exactly what you will get; for example if you enter C5 you know you will get a green Gatorade. However, if you approach the vending machine wanting a Pepsi, then there are several options you could enter to get one; A1, A2, A3, A4, or A5.

In this example, the relationship that inputs the location of the drink you request (such as C5) and outputs the drink you get as a result (green Gatorade) would be a function. In contrast, the relation whose input is what drink you want (such as Pepsi) and outputs the location you must enter to get that drink (A1 through A5) would not be a function, because there are multiple outputs for the same input.

Note that it is perfectly natural to ask what drink is in a given location, as well as asking what location you should type in for a particular drink you want. Both of these situations are perfectly natural, and yet one is a function and one is not.

1 : What is meant by context, with regards to mathematical relations?

The context required to define a function has a special set of terminology in mathematics; the domain, codomain, and range, which we discuss in the next section.

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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