A function maps a set of inputs onto permissible outputs, and each
input maps onto one and only one output.
LEARNING OBJECTIVES [ edit ]
List the characteristics of a function
Calculate the value of the output of a function given its input
KEY POINTS [ edit ]
Functions are relations between a set of inputs and a set of permissible outputs with the property
that each input is related to exactly one output.
Typically functions are named with a single letter, most commonly f, g and h. A function takes the
form f(x) for one input variable, but it can take any number of variables, e.g. f(x, y, z).
Functions can be thought of as a machine in a box open on two ends. You put something in one
end, something happens to it in the middle, and something pops out the other end.
TERMS [ edit ]
relation
A relation is a connection between numbers in one set and numbers in another.
function
A relation in which each element of the input is associated with exactly one element of the output.
output
data sent out of the computer, as to output device such as a monitor or printer
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Functions
What Are Functions?
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 x2.
Functions are typically named with a
single letter, most typically f, so we'll call
this function f. The output of a function f
corresponding to an input x and is
denoted by f(x) (read "f of x"). In this
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example, if the input is −3, then the output is 9, and we may write f(−3) = 9. The input
variable(s) are sometimes referred to as the argument(s) of the function.
In the case of a function with just one input variable, the input and output of the function can
be expressed as anordered pair, ordered so that the first element is the input, the second the
output. In the example above, f(x) = x2, we have the ordered pair (−3, 9). If both the input
and output are real numbers, this ordered pair can be viewed as the Cartesian coordinates of
a point on the graph of the function.
Another commonly used notation for a function is f:X→Y, which reads as saying that f is a
function that maps values from the set X onto values of the set Y.
Functions As a Box
Functions are often described as a machine in a box open on two ends. You put something in
one end, something happens to it in the middle, and something pops out the other end. The
function is the machine inside, and it's defined by what it does to whatever you give it .
INPUT x
FUNCTION f:
OUTPUT f(x)
Function Machine
A function f takes an input x and returns an output f(x). One metaphor describes the function as a
"machine" or "black box" that for each input returns a corresponding output.
Let's say the machine has a blade that slices whatever you put into it in two and sends one
half out the other end. If you put in a banana, you'd get back half a banana. If you put in an
apple, you'd get back half an apple .
Fruit Halving Function
This shows a function that takes a fruit as input and releases half the fruit as output.
You may wonder what happened to the other half of the piece of fruit, but since this is
algebra, the things that go in and come out of functions will be numbers, so the box simply
fills up with numbers and will not break. Let's define the function to take what you give it
and cut it in half, that is, divide it by two. If you put in 2, you'd get back 1. If you put in 57,
you'd get back 28.5. The function machine allows us to alterexpressions. In this example, f(2) = 1 f(57) = 28.5 f(x) =
1
2
x .
Functions As a Relation
Functions can also be thought of as a subset of relations. A relation is a connection between
numbers in one set and numbers in another . In other words, each number you put in is
associated with each number you get out. The difference is that in a function, every input
number is associated with exactly one output number, whereas in a relation, an input
number may be associated with multiple or no output numbers. This is an important fact
about functions that cannot be stressed enough: every possible input to the function must
have one and only one output. All functions are relations, but not all relations are functions.
Mapping of a Function
The oval on the left is the domain of the function f, and the oval on the right is the range.