relations and functions another foundational concept copyright © 2014 – curt hill

Relations and Functions

Another Foundational Concept

Copyright © 2014 – Curt Hill

Relations• Ordered pairs express relationships• A set of ordered pairs define a

relation• This relation may or may not be a

function• Every function is a relation but not

every relation is a function


Example• An example relation is the

origin and destination of all airline flights– (Fargo, Bismarck)– (Fargo, Minneapolis)– (Fargo, Winnipeg)– (Minneapolis, Fargo)– (Minneapolis, St. Louis)

• When we see a set of ordered pairs, they are always a relation


Terminology• The first item in the ordered

pair is the domain and second is the range– Rosen likes codomain instead of

range• The domain and range are sets• Airline example the domain

was {Fargo, Minneapolis} and the range {Bismarck, Minneapolis, Winnipeg, St. Louis} Copyright © 2014 – Curt Hill

Example• Suppose the following ordered

pairs:• (2,3),(2,4),(3,5),(4,2),(7,4)• What is the domain and range?


The concept of a function• A function is a relation with

one item in the range corresponding to only one item in the domain

• A function is like a black box:– You push a value into the black

box– Turn the crank – Out comes a new number

• Example square root• Push in 1 out comes 1• Push in 4 out comes 2• Push in 9 out comes 3• Push in 16 out comes 4


Airlines Again• The airline example was not a

function• Fargo flew to both Minneapolis and

Bismarck• A function must always give the

same output for the same input• A function may have many

parameters– If we increase the parameters from

origination city to also include flight number it will then be a function


Functions• Also called mappings or

transformations– A function maps one or more values

onto a single value– It also transforms values

• Most sets of ordered pairs are the result of a function

• The parameter is the first item and the result is the second

• So the square root function produces the following ordered pairs (1,1) (4,2) (9,3) (16,4)


Again• Many functions produce an

infinite number of ordered pairs

• We have two values here:– The independent and dependent

variable• Independent is the one that is

input to the function– There may multiple independents

• The dependent variable is output Copyright © 2014 – Curt Hill

Notation• A function may be represented

as:y = f(x)

• When we know the exact computation of f we can also write– f(x) = 2x+3

• We can also use set notation to represent a function: {(x,y)| y=3x-2}

• We traditionally use the letters starting at f to designate functions: f,g,h, F,G,H


Equality• Two functions are equal if and only

if they have the same:– Domain– Range– Mapping


Square• Consider the square function:

– f(x) = x2

– Domain: all reals– Range: positive reals and zero

• Notice that this maps two values onto a single value– f(2)=f(-2)=4

• Still a function since each value maps to just one


Other functions• Contrast the square function with:

– f(x) = 2x+1– Domain and range is all reals

• This function has two properties that the square function does not:– One to one– Onto

• One to one means it does not map two values onto one

• Onto means that the domain and range are exactly the same


Terminology• One to one functions are also

called injunctions– Adjective form is injective

• Onto functions are also called surjections– Adjective is surjective

• If both one to one and onto it is a bijection– Bijective


Composite functions• We can also do composite

functions, which is f(g(x)) or (fg)(x)

• Suppose f(x) = 4x – 3• Suppose g(x) = 2x + 1• Then 2x+1 gets substituted for

x in f:• Then (f g)(x) = 4(2x+1)-3 =

8x+1• All this is having a function as an

input to another functionCopyright © 2014 – Curt Hill

Inverse Functions• Functions that are one to one allow

the possibility of an inverse function

• An inverse function reverses the mapping, range and domain of a function

• The inverse of f(x) is denoted f-1(x)• f(f-1(x)) = f-1(f(x)) = x• The inverse function only exists for

one to one functionsCopyright © 2014 – Curt Hill

Graphs• In this section a graph has nothing

to do with visual representation• Instead it is a set of ordered pairs

– Which is actually how we generate the visual representation that you remember from algebra

• Let us consider a couple of examples


Graph Example 1• Suppose the domain of function f is

the set S = {-1, 0, 2, 4, 7}• If f(x) = 2x + 1, what is the graph?• It is a set of ordered pairs, the first

item comes from S and the second is found by plugging the item into f

• {(-1,-1), (0,1), (2,5), (4,9), (7,15)}


Graph Example 2• The above method does not work

that well for very large sets• We do this using set builder

notation• Specify a set of ordered pairs and

give the formula for generation• If the domain of the previous

function was the real numbers then we would give the following:{(x,y)|xR y=f(x)}


Important Functions• Floor of x is denoted by

– The largest integer smaller than x• Ceiling of x is denoted by

– The smallest integer larger than x• Factorial of a positive integer n is

n!– n! = 2*3*…*n


Partial functions• We sometimes refer to functions

with limited domains as partial functions

• The square root function is not the inverse of the square function– It’s domain is reals greater than or

equal zero• Using an element outside of the

domain is undefined


Exercises• 2.3

– 5, 9, 21, 27 61, 77


relations and functions another foundational concept copyright © 2014 – curt hill

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