In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number "x" to its square "x". The output of a function "f" corresponding to an input "x" is denoted by "f"("x") (read ""f" of "x""). In this example, if the input is −3, then the output is 9, and we may write . Likewise, if the input is 3, then the output is also 9, and we may write . (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function. 

Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.

Now, provide a numbered list of answers to these questions:

1. What are functions?
2. Between what?
3. What are different types of function in today's mathematics?
4. What sort of image could you use to describe them?
5. How could a function's representation be implied?
6. Any other way?
7. What way is that?
8. How could a function be directly defined?
9. Can each input give many outputs?
10. Can an output give many inputs?

Numbered answers:
1. relations
2. inputs and outputs
3. the central objects of investigation
4. a graph
5. as the inverse to another function
6. yes
7. a solution of a differential equation
8. by a table
9. no
10. yes