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**A-Level Maths: Core 3 for Edexcel**

C3.2 Algebra and functions 2 This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation. 1 of 64 © Boardworks Ltd 2006

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Mappings A mapping is made up of two sets and a rule that relates elements from the first set, the input set, to the elements of the second set, the output set (or image set). Rule Input set Output set DOMAIN RANGE The set of all permissible inputs is called the domain of the mapping. The set of all corresponding outputs is called the range of the mapping. It is possible for the set we are mapping onto to contain elements that are not in the range of the mapping. This entire set, that includes the range as a subset, is called the co-domain. However, knowledge of the co-domain is not needed at this stage. For example, suppose we have the input set A = {1, 2, 3, 4, 5}. This set represents the domain of the mapping.

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Mappings This set is related to the output set B = {–1, 1, 3, 5, 7} by the rule “double and subtract 3”. Double and subtract 3 1 2 3 4 5 DOMAIN –1 7 RANGE A B –1 1 3 5 7 “m : a → 2a – 3” can be read as “m is the rule such that each element a is mapped onto 2 times a minus 3”. If we call this rule m we can write it using mapping notation as: m: a → 2a – 3 Where a represents the elements in set A.

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**Types of mappings A mapping can be described as: One-to-one**

1 2 3 4 –1 Subtract 2 One-to-one In a one-to-one mapping, each element in the domain is mapped onto exactly one element in the range. For example: 1 4 9 –1 –2 2 –3 3 Square Many-to-one In a many-to-one mapping, two or more elements in the domain can be mapped onto the same element in the range. For example: A non-numerical example of a one-to-one mapping could be the mapping between each member of the class and their position in the register. Each element in the domain (each person in the class) is mapped onto a unique element in the range (one position in the register). A similar non-numerical example of a many-to-one mapping could be the mapping between each member of the class and their favourite colour. Each element in the domain (each person in the class) is mapped onto one element in the range (a colour). However, in this case it is most likely that many people will share the same favourite colour so that some elements in the range will have several elements mapped onto them. These two types of mapping can be classified as functions.

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**Types of mappings One-to-many**

–1 1 –2 2 –3 3 Square root 4 9 In a one-to-many mapping each element in the domain can be mapped onto two or more elements in the range. For example: 5 6 7 8 1 2 4 Is a factor of Many-to-many In a many-to-many mapping more than one element in the domain can be mapped onto more than one element in the range. For example: A non-numerical example of a one-to-many mapping could be the mapping between each member of the class and their brothers and sisters. Each element in the domain (each person in the class) can mapped onto several elements in the range (the students’ brothers and sisters). However, if two members of the class happen to be siblings this relationship will be many-to-many. Another non-numerical example of a many-to-many mapping could be the mapping between each member of the class and the subjects they are studying at A-level. Each element in the domain (each person in the class) will be mapped onto several elements in the range (all the A-level subjects) and many elements in the range will have many elements from the domain mapped onto them. These two types of mappings are not functions.

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Functions A function is a special type of mapping such that each member of the domain is mapped to one, and only one, element in the range. In other words, Only a one-to-one or a many-to-one mapping can be called a function. One-to-many and many-to-many mappings are not functions. Most functions you will meet are defined over a continuous domain, such as the set of all real numbers. Such functions are best represented by a graph plotting the elements in the domain along the horizontal axis against the corresponding members of the range along the vertical axis.

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**Which graphs represent functions?**

For each graph drag the blue and red horizontal and vertical lines through it to determine whether it shows a one-to-one, many-to-one, one-to-many or many-to-many relationship.

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Function notation We usually use the letter f to represent a function but other letters such as g and h can also be used. The letter x is normally used to represent elements of the domain (input values). Since we can choose the value of x it is called an independent variable. For example, if we have the function “square and add 5” this can be written as f of x equals x squared plus 5 f(x) = x2 + 5 We can also use mapping notation to write f: x → x2 + 5 f is a function such that x is mapped onto x squared plus 5.

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**Function notation If we choose x to be –3, say, we can write**

= 14 We say that 14 is the image of –3 under the function f. The letter y is normally used to represent elements of the range (output values), so y = f(x). Since the value of y is determined by the function acting on x it is called a dependent variable. Since the domain of f(x) = x2 + 5 has not been given, we assume that x belongs to the set of real numbers, Discuss the fact that although x can be any real number, the corresponding value of f(x) will always be greater than 5. This is because the least value that x2 can take is 0. What is the range of this function? f(x) cannot be less than 5, so the range is: f(x) ≥ 5

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**Finding the range of a function**

Functions and mappings Finding the range of a function Composite functions Inverse functions and their graphs The modulus function Transforming functions Examination-style questions Contents 10 of 64 © Boardworks Ltd 2006

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**The domain and range of a function**

Remember, The domain of a function is the set of values to which the function can be applied. The range of a function is the set of all possible output values. A function is only fully defined if we are given both: the rule that defines the function, for example f(x) = x – 4. the domain of the function, for example the set {1, 2, 3, 4}. Given the rule f(x) = x – 4 and the domain {1, 2, 3, 4} we can find the range: {–3, –2, –1, 0}

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**The domain and range of a function**

It is more common for a function to be defined over a continuous interval, rather than a set of discrete values. For example: The function f(x) = 4x – 7 is defined over the domain –2 ≤ x < 5. Find the range of this function. Since this is a linear function, substitute the smallest and largest values of x: When x = –2, f(x) = –8 – 7 = –15 With many-to-one functions such as f(x) = sin(x) or f(x) = x2, substituting the smallest and largest values of x may not give the range of the function. In these cases it is more important to sketch the function. When x = 5, f(x) = 20 – 7 = 13 The range of the function is therefore –15 ≤ f(x) < 13

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