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© Boardworks Ltd of 64 © Boardworks Ltd of 64 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.

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© Boardworks Ltd of 64 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). Input set Output set Rule 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. For example, suppose we have the input set A = {1, 2, 3, 4, 5}. This set represents the domain of the mapping. D OMAIN R ANGE

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© Boardworks Ltd of 64 Double and subtract D OMAIN – – R ANGE AB Mappings This set is related to the output set B = {–1, 1, 3, 5, 7} by the rule double and subtract 3. If we call this rule m we can write it using mapping notation as: m : a 2 a – 3 Where a represents the elements in set A. –

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© Boardworks Ltd of –1 1 –2 2 –3 3 –1 1 –2 2 –3 3 Square Types of mappings A mapping can be described as: 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: 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: – – Subtract 2

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© Boardworks Ltd of Is a factor of –1 1 –2 2 –3 3 –1 1 –2 2 –3 3 Square root Types of mappings One-to-many In a one-to-many mapping each element in the domain can be mapped onto two or more elements in the range. For example: 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:

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© Boardworks Ltd of 64 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, 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. Only a one-to-one or a many-to-one mapping can be called a function.

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© Boardworks Ltd of 64 Which graphs represent functions?

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© Boardworks Ltd of 64 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). For example, if we have the function square and add 5 this can be written as We can also use mapping notation to write Since we can choose the value of x it is called an independent variable. f ( x ) = x f : x x f of x equals x squared plus 5 f is a function such that x is mapped onto x squared plus 5.

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© Boardworks Ltd of 64 Function notation 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. If we choose x to be –3, say, we can write f (–3) = (–3) 2 + 5= 14 We say that 14 is the image of –3 under the function f. Since the domain of f ( x ) = x has not been given, we assume that x belongs to the set of real numbers, 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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© Boardworks Ltd of 64 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 © Boardworks Ltd of 64 Finding the range of a function

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© Boardworks Ltd of 64 The domain and range of a function Remember, 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} 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.

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© Boardworks Ltd of 64 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: When x = –2, f ( x ) = The range of the function is therefore –8 – 7 =–15 When x = 5, f ( x ) = 20 – 7 =13 Since this is a linear function, substitute the smallest and largest values of x : –15 f ( x ) < 13 The function f ( x ) = 4 x – 7 is defined over the domain –2 x < 5. Find the range of this function.

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