2A 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.
3x 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.
4A 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
5What 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.
6The 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.
7The 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
8e.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
9BUT 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 )
11Asymptotes: 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
12Vertical 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
13SUMMARYTo 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
14Exercise1. Sketch the functions whereFor each function write down the domain and rangeSolution:(a)(b)domain:Rdomain:Rrange:Rrange:
15x + 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
16One-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.
17One-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
19Other 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.
20We’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.