Question

Which set of coordinates represents a function?

Answer 1

Answer:

Assuming that the domain is $\lbrace0,\, 1,\, 2,\, 3\rbrace$ (the set of whole numbers between 0 and 3, inclusive,) then only the third choice would represent a function:

$\lbrace{(2,\, 15),\, (3,\, -3),\, (1,\, 3),\, (0,\, 0)\rbrace}$.

Step-by-step explanation:

A function needs to be

• one-to-one for all inputs in its domain, and
• defined for all values on its domain.

A map is one-to-one only if it maps each input to only one output.

For example, in $(0,\, 5)$ in the first choice, $0$ is the input while $5$ is the output. If this set of coordinates is indeed one-to-one, then this input,

Similarly, the second set of coordinates tries to map the input $1$ to both $10$ and $2$. This option isn't one-to-one and does not represent a function, either.

Indeed, the third, fourth, and fifth options are all one-to-one. However, there are gaps in the domain of the fourth and fifth option. (The domain of a map is the set of all the points in its definition.)

• The domain of the third option is $\lbrace0,\, 1,\, 2,\, 3\rbrace$.
• The domain of the fourth option is $\lbrace-1,\, 1,\, 3,\, 5\rbrace$.
• The domain of the fifth option is $\lbrace-2,\, 1,\, 4,\, 5\rbrace$.

Note, that whether these options are functions depend on the choice of the domain. For example, if the domain is merely $\lbrace1\rbrace$, then all three options would be functions. However, if the domain is the set of all integers ($\mathbb{Z}= \lbrace\dots,\, -2,\, -1,\, 0,\, 1,\, 2,\, \dots\rbrace$,) then none of these choices will be a function.