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1: Functions “Teach A Level Maths” Vol. 2: A2 Core Modules

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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

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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.

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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

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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.

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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

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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.

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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.

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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 )

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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 =

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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

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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

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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

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Functions (b) (a) Exercise For each function write down the domain and range 1.Sketch the functions where Solution: range: domain: range:

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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

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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

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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

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Functions Other one-to-one functions are: and

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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.

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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.

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