# Functions

As used in ordinary language, the word function indicates
dependence of a varying quantity on

another. If I tell you that your grade in this class is a function of your
overall average, you

interpret this to mean that I have a rule for translating a number in the range
of 0 to 100 into a

letter grade. More generally, suppose two sets of objects are given: set A and
set B; and

suppose that with each element of A there is associated a particular element of
B. These three

things: the two sets and the correspondence between elements comprise a
function.

A function f is a mapping from a set D to
a set T with the property that for each element d inD, f maps d to a single element of T, denoted f(d). Here D is called the domain of f, and T iscalled the target or co-domain. We write f: D → T. We also
say that f(d) is the image of dunder f, and we call the set of all images the range R of f. |

A mapping might fail to be a function if it is not defined
at every element of the domain, or if

it maps an element of the domain to two or more elements in the range:

Consider the mappings shown in the diagram above. We note
that examples **a** and **b** are both

functions since every element in the first set (the domain) maps to a single
element in the second

set (the co-domain or target). Note that it is fine for two elements in the
domain to map to the

same element in the co-domain (as is the case in **b**). We point out that
example c is not a function,

since there is an element in the domain which does not map to any element in the
co-domain.

Also, example **d** is not a function since there is an element in the domain
that maps to more than

one element in the co-domain.

One way to define a function is to provide a table that shows the mapping for
each element of the

domain. For example, in a small class we might have

the set of students S = {Maria, Clara. Tom, Dick, Harry}

the set of possible grades G = {A, B, C, D, NP}

One possible function f: S → G would be:

d | f(d) |

Maria Clara Tom Dick Harry |
A B C A B |

The table completely defines the function by showing every
mapping. Note that all the

possible grades are not used, i.e., the range is not the same as the co-domain.
It is still

convenient to call this a function from S to G in order to indicate the
possibilities.

The following table does not define a function from S to G, because not every
member of S is

mapped to a grade:

d | f(d) |

Maria Clara Tom Dick Harry |
A B C B |

The following table does not define a function from S to
G, because one member of S is

mapped to two grades :

d | f(d) |

Maria Clara Tom Dick Harry Harry |
A B C A B NP |

Another way to define a function to specify a rule for how
the function operates , rather than listing

out the mapping. For example, using the common notations

N: the set of natural numbers {1, 2, 3, ...}

Z: the set of all integers {..., -2, -1, 0, 1, 2, ...}

we could define a function f : N → N with the rule f (a) = 2a. The specification
of the domain and

co-domain are considered to be part of definition, so the function g : Z → Z
where g(a) = 2a is not

the same function as f, even though the rules are the same .

Note that as in the example above, the range of these functions is not the same
as the co-domain.

For example, the range of f is positive even integers, which is not the same as
N. We write

f : N → N to convey the information that everything in the range of f is a
natural number , without

commenting (until we give the rule) on whether every natural number is actually
in the range.

**Types of Functions
One-to-One Functions**

A function is called one-to-one, or
injective, if it maps distinct elements of the domain to distinctelements of the range. More formally, a function f is one-to-one
if and only if x ≠ y implies thatfor f(x) ≠ f(y) for all x and y in the domain of f. Another way to say this is that f(x) = f(y) implies x = y. A function is said to be an injection if it is one-to-one. |

For example, the function f: N → N defined by the rule f (x)
= x^{2 }is an injection. However,

the function g: Z → Z defined by the same rule is not an injection, since g(-x)
and g(x) both

map to x^{2}.

**Onto Functions**

A function f: A →B is called onto, or
surjective,
if and only if for every element b in B there is atleast one element a in A that maps to it, i.e., there is at least one element a such that f(a) = b. Another way to say this is that the range of f is equal to its target. A function is said to be a surjection if it is onto |

For example, the function f(x) = x + 1 from the set of
integers to the set of integers is onto, since

for every integer y there is an integer x such that f(x) = y. By definition, f(x)
= y if and only if

x + 1 = y, implying that x = y – 1. So, for any y, we can find a value x so that
f(x) = y holds.

The function f(x) = x^{2 }from the set of integers to the set of integers is not
onto. This follows from

the fact that there is no integer x such that f(x) = x^{2 }= –1.

**One-to-one Correspondence Functions**

A function f is a one-to-one correspondence, or
bijective, if it is both one-to-one and onto. Sucha function is called a bijection. |

The function f(x) = x + 1 from the set of integers to the
set of integers is a bijection. It is one-to-

one since x + 1 = y + 1 if and only if x = y. And as we saw above, the function
is onto.

As examples, consider the four functions shown below:

Function **a** is one-to-one and onto (it is a bijection).

Function **b** is one-to-one, but not onto.

Function **c** is not one-to-one, but is onto.

Function **d **is neither one-to-one nor onto.

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