2 A function is a rule , which calculates values of for a set of values of x.e.g and are functions.is called the image of xAnother Notationmeansis often replaced by y.
3 x x We can illustrate a function with a diagram . -1 -3 . -1 . 1.5 2 . .2.11.5-3.xx-1.2.A few of the possible values of x3.2.The rule is sometimes called a mapping.
4 A bit more jargonTo define a function fully, we need to know the values of x that can be used.The set of values of x for which the function is defined is called the domain.In the function any value can be substituted for x, so the domain consists ofall real values of xWe writeRWe say “ real ” values because there is a branch of mathematics which deals with numbers that are not real.stands for the set of all real numbersRmeans “ belongs to the set ”So, means x belongs to the set of real numbersR
5 What is a real number?The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as p and the 2;or, a real number can be given by an infinite decimal representation, such as , where the digits continue in some way;or, the real numbers may be thought of as points on an infinitely long number line.Real numbers measure continuous quantitiessuch as weight, height, speed, time.
6 The range of a function is the set of values given by . e.g. Any value of x substituted into gives a positive ( or zero ) value.So the range of isIf , the range consists of the set ofy-values, sodomain: x-valuesrange: y-valuesTip: To help remember which is the domain and which the range, notice that d comes before r in the alphabet and x comes before y.
7 The set of values of x for which the function is defined is called the domain. The range of a function is the set of values given by the rule.domain: x-valuesrange: y-values
8 e.g. 1 Sketch the function where and write down its domain and range.Solution: The quickest way to sketch this quadratic function is to find its vertex by differentiating.so the vertex is
9 BUT we could have drawn the sketch for any values of x. So, the graph of isxdomain:The x-values on the part of the graph we’ve sketched go from -5 toBUT we could have drawn the sketch for any values of x.So, we get( x is any real number )RBUT there are no y-values less than -5, . . .so the range is( y is any real number greater than, or equal to, -5 )
10 ) ( x f y = e.g.2 Sketch the function where . Hence find the domain and range ofSolution: is a translation from of)(xfy=so the graph is:domain: x-valuesrange: y-values( We could write instead of y )
11 Asymptotes: if a function approaches a particular x or y value then the equation of this value is called an asymptote.Notice how each value of x maps (links) to ONLY one value of y .This is the definition of a function.If each value of x maps to MORE than one value of y it is NOT a function
12 Vertical asymptote at x = 2 As you cannot divide by zerox – 2 0Domain : x 2Vertical asymptote at x = 2The function approaches the line x = 2 but never meets itHorizontal asymptote at y = 3.The graph approaches the line y = 3 but never meets it.Range : y 3 or y<3 and y>3
13 SUMMARYTo define a function we needa rule and a set of values.Notation:meansFor ,the x-values form the domainthe or y-values form the rangee.g. For ,the domain isthe range is orR
14 Exercise1. Sketch the functions whereFor each function write down the domain and rangeSolution:(a)(b)domain:Rdomain:Rrange:Rrange:
15 x + 3 must be greater than or equal to zero. We can sometimes spot the domain and range of a function without a sketch.e.g. For we notice that we can’t square root a negative number ( at least not if we want a real number answer ) so,x + 3 must be greater than or equal to zero.So, the domain isThe smallest value of is zero.Other values are greater than zero.So, the range is
16 One-to-one and many-to-one functions Consider the following graphsandEach value of x maps to only one value of y . . .Each value of x maps to only one value of y . . .and each y is mapped from only one x.BUT many other x values map to that y.
17 One-to-one and many-to-one functions Consider the following graphsandis an example of a many-to-one functionis an example of a one-to-one function
19 Other many-to-one functions are: Here the many-to-one function is two-to-one( except at one point ! )This is a many-to-one function even though it is one-to-one in some parts.It’s always called many-to-one.
20 We’ve had one-to-one and many-to-one functions, so what about one-to-many? One-to-many relationships do exist BUT, by definition, these are not functions.e.g.is one-to-many since it gives 2 values of y for all x values greater than 1.This is not a function.Functions cannot be one-to-many.So, for a function, we are sure of the y-value for each value of x. Here we are not sure.