Functions e.g. and are functions. A function is a rule, which calculates values of for a set of values of x. is often replaced by y. Another Notation means is called the image of x
Functions x x -- A few of the possible values of x -- -- We can illustrate a function with a diagram The rule is sometimes called a mapping.
Functions We say “ real ” values because there is a branch of mathematics which deals with numbers that are not real. A bit more jargon To 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 of all real values of x means “ belongs to the set ” So, means x belongs to the set of real numbers stands for the set of all real numbers We write
Functions What is a real number? The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as 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 quantities such as weight, height, speed, time.
Functions Tip: 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. If, the range consists of the set of y-values, so e.g. Any value of x substituted into gives a positive ( or zero ) value. So the range of is The range of a function is the set of values given by. domain: x -values range: y - values
Functions The range of a function is the set of values given by the rule. domain: x - values range: y - values The set of values of x for which the function is defined is called the domain.
Functions Solution: The quickest way to sketch this quadratic function is to find its vertex by differentiating. e.g. 1 Sketch the function where and write down its domain and range. so the vertex is.
Functions so the range isSo, the graph of is The x-values on the part of the graph we’ve sketched go from - to + ... BUT we could have drawn the sketch for any values of x. ( y is any real number greater than, or equal to, - ) BUT there are no y -values less than - ,... x domain: So, we get ( x is any real number )
Functions domain: x - values range: y - values e.g.2 Sketch the function where. Hence find the domain and range of. so the graph is: ( We could write instead of y ) Solution: is a translation from of xfy =
Functions 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
Functions As you cannot divide by zero x – 2 0 Domain : x 2 Vertical asymptote at x = 2 The function approaches the line x = 2 but never meets it Horizontal asymptote at y = 3. The graph approaches the line y = 3 but never meets it. Range : y 3 or y 3
Functions SUMMARY To define a function we need a rule and a set of values. For, the x -values form the domain Notation: means the or y -values form the range e.g. For, the domain is the range is or
Functions (b) (a) Exercise For each function write down the domain and range 1.Sketch the functions where Solution: range: domain: range:
Functions So, the domain is 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 must be greater than or equal to zero. The smallest value of is zero. Other values are greater than zero. So, the range is
Functions One-to-one and many-to-one functions Each value of x maps to only one value of y... Consider the following graphs Each value of x maps to only one value of y... BUT many other x values map to that y. and each y is mapped from only one x. and
Functions One-to-one and many-to-one functions is an example of a one-to-one function is an example of a many-to-one function Consider the following graphs and
Functions Other one-to-one functions are: and
Functions Here the many-to-one function is two-to-one ( except at one point ! ) Other many-to-one functions are: This is a many-to-one function even though it is one-to-one in some parts. It’s always called many- to-one.
Functions This is not a function. Functions cannot be one-to-many. 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. is one-to-many since it gives 2 values of y for all x values greater than 1. e.g. So, for a function, we are sure of the y -value for each value of x. Here we are not sure.