Question

Explain what a function is and how to determine whether tables, graphs, equations, and ordered pairs are a function.

Answer 1

In order to define a function, you need two sets, say
`A` and
`B`.
`A` will be the domain, i.e. the "starting" set.
`B` will be the range (or co-domain), and you can think of this as the "landing" set.

A function is a rule that maps, following the same logic, every element of
`A` into one and only one elment of
`B`.

Most of the time, you'll deal with numeric function, i.e. functions that take a number as input and give another number as output. For example, writing

`f(x)=2x`

means that your function takes a certain number
`x` as input, and return twice that number,
`2x`, as output.

Finally, a function can also be described by explicit pair of inputs and outputs. For example, defining a function
`f` with the table

`\begin{array}c1&2\\4&6\\7&89\end{array}`

means that
`f` maps 1 onto 2, 4 onto 6 and 7 onto 89. Such a collection of pairs is actually a function if:

• Every point in the domain is actually associated with an output in the range (1)
• Every point in the domain is associated with only one output in the range (2)

(1) for example, if your domain is
`A=\{1,2,3,4\}`, and the range is
`B=\{13, 14, 55, 59\}`, the table

`\begin{array}c1&13\\2&14\\3&55\end{array}`

does not represent a function from
`A` to
`B`, because 4 is in the domain, but it is not mapped into anything. Similarly, the table

`\begin{array}c1&13\\2&14\\3&55\\4&89\end{array}`

is not a valid function, because 4 is mapped onto 89, which is not in the range.

(2) for example, using the same sets as above, the table

`\begin{array}c1&13\\1&14\\2&14\\3&55\\4&59\end{array}`

does not represent a function from
`A` to
`B`, because 1 is mapped into both 13 and 14, but a value in the domain can be mapped into only one value in the range.