Question

How do you define a function?

Answer 1

The definition for a function describes the relationship between objects in the 'domain' of the function and objects in its 'range'.

See explanation...

Explanation:

The definition for a function describes the relationship between objects in the 'domain' of the function and objects in its 'range' in such a way that for any object in the domain there is a unique corresponding object in the range.

The definition of a function can take several forms. Here are some popular ones:

Formula

A function definition can be a formula which given a value allows you to compute another value. For example:

`f(x) = 2x+3`

is a function definition which given a number `x` tells you that you can compute `f(x)` by multiplying `x` by two then add three.

This example function definition works as a definition of a function from integers to integers, rational numbers to rational numbers or real numbers to real numbers.

Explicit mapping

A function definition can tell you explicitly what maps to what. This may be expressed as a set of ordered pairs.

For example, the set `{ (1, 1), (2, 4), (3, 9), (4, 16) }` can be viewed as the definition of a function from the set `{ 1,2,3,4 }` to the set `{ 1, 4, 9, 16 }`, which maps `1` to `1`, `2` to `4`, etc.

Miscellaneous

For example, define a function `p:NN->NN` where `p(n) =` the `n`th prime number. So `p(1) = 2`, `p(2) = 3`, etc.

This is obviously a consistent definition of a function, but there is no formula for the `n`th prime number.