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 Copyright © 2014 – Curt Hill

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 Copyright © 2014 – Curt Hill

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? Copyright © 2014 – Curt Hill

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 Copyright © 2014 – Curt Hill

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 Copyright © 2014 – Curt Hill

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) Copyright © 2014 – Curt Hill

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 Copyright © 2014 – Curt Hill

Equality Two functions are equal if and only if they have the same: –Domain –Range –Mapping Copyright © 2014 – Curt Hill

Square Consider the square function: –f(x) = x 2 –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 Copyright © 2014 – Curt Hill

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 Copyright © 2014 – Curt Hill

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 Copyright © 2014 – Curt Hill

Composite functions We can also do composite functions, which is f(g(x)) or (f  g)(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 function Copyright © 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 functions Copyright © 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 Copyright © 2014 – Curt Hill

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)} Copyright © 2014 – Curt Hill

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)|x  R  y=f(x)} Copyright © 2014 – Curt Hill

Important Functions Copyright © 2014 – Curt Hill

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 Copyright © 2014 – Curt Hill

Exercises 2.3 – 5, 9, 21, 27 61, 77 Copyright © 2014 – Curt Hill