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Introduction • This chapter focuses on functions • You will learn how to answer questions involving single and multiple functions • You will also see how to decide whether an expression is a function or not Functions Mapping Diagrams A mapping diagram transforms one set of numbers into a different set of numbers. It can be described in words or using algebra. They can also be represented by a Cartesian graph. Add 3 onto the set {-3, 1, 4, 6, x) Set A Set B -3 1 4 6 x 0 4 7 9 x+3 y=x+3 y The original numbers (Set A, or ‘x’) are known as the domain. 8 6 The results (Set B, or ‘y’) are known as the range (ie range of answers) 4 2 -2 2 4 6 x 2A Functions Mapping Diagrams A mapping diagram transforms one set of numbers into a different set of numbers. It can be described in words or using algebra. They can also be represented by a Cartesian graph. Square the set {-1, 1, -2, 2, x) Set A Set B -1 1 -2 2 x 1 4 x2 y The original numbers (Set A, or ‘x’) are known as the domain. y = x2 8 6 The results (Set B, or ‘y’) are known as the range (ie range of answers) 4 2 -2 2 4 6 x 2A Functions Functions A function is a mapping whereby every element in the domain is mapped to only 1 element in the range. ie) Whatever number you start with, there is only 1 possible answer to the operation performed on it. An example of a mapping which is not a function would be square rooting, where the starting number may result in no answer, or 2 answers. One-to-one Function eg) f(x) = x + 5 Set A Set B eg) f(x) = 3x - 2 Many-to-one Function Set A Set B eg) f(x) = x2 + 1 eg) f(x) = 6 - 3x2 Not a function Set A Set B eg) f(x) = √x eg) f(x) = 1/x 2B Functions Functions A function is a mapping whereby every element in the domain is mapped to only 1 element in the range. One-to-one Function ‘A value in the domain (x) gets mapped to one value in the range’ Many-to-one Function ie) Whatever number you start with, there is only 1 ‘Multiple values in the domain (x) get possible answer to the operation performed on mapped to the same it. value in the range’ An example of a mapping which is not a function would be square rooting, where the starting number may result in no answer, or 2 answers. ‘A value in the range can be mapped to none, one or more values in the range’ Not a function 2B Functions Example Question Given that the function g(x) = 2x2 + 3, find; g(x) = 2x2 + 3 a) g(3) = 2(3)2 + 3 = 2(9) + 3 = 21 a) the value of g(3) b) the value(s) of a such that g(a) = 35 b) g(a) = 35 2a2 + 3 = 35 2a2 = 32 a2 = 16 a =±4 2B Functions Example Question Given that the function g(x) = 2x2 + 3, find; a) the value of g(3) b) the value(s) of a such that g(a) = 35 c) the range of the function g(x) ≥ 3 y = 2x2 + 3 g(x) You can get any value bigger than, or including 3… 8 6 4 2 -2 -1 1 2 x To work out the range of the function; - Sketch it first - the range is the set of answers you get (ie the ‘y’ values – now labelled as g(x)…) - Use an Inequality if there is a continuous set of values 2B Functions g(x) x An important bit of notation to remember… x can be any ‘real number’ This is for the domain g(x) can be any ‘real number’ This is for the range Real Number: A number which has a place on a normal number line. Includes positives, negatives, roots, pi etc… Does not include imaginary numbers – eg √-1 2B Real Number: A number which has a place on a normal number line. Includes positives, negatives, roots, pi etc… Functions Domain changes y A mapping which is not a function, can be made into one by changing/restricting the domain (the starting values) x eg) y = +√x This will not be a function as some values in the domain (x) will not give an answer in the range (y). For example, -2 If we restrict the domain to x ≥ 0, then all values in the domain will map to one value in the range. f ( x) x x R, x 0 The function x is real numbers x is greater than 0 It now therefore meets the criteria for being a function! 2C Functions Find the range of the following function, and state if it is one-toone or many-to-one. f(x) = 3x – 2, domain {x = 1, 2, 3, 4} f(x) = 3x – 2, {x = 1, 2, 3, 4} Domain Range 1 1 2 4 3 7 4 10 No inequality used as there are only certain values (discrete) Range of f(x): {1, 4, 7, 10} Description: One to One 2C Functions g(x) Find the range of the following function, and state if it is one-toone or many-to-one. 20 15 Range g(x) = x2, domain {x є R, -5 ≤ x ≤ 5} g(x) = x2 10 5 g(x) = x2, {-5 ≤ x ≤ 5} Inequality, so you will have to sketch the graph Range of g(x): 0 ≤ g(x) ≤ 25 Description: Many to one -4 -2 2 4 x Inequality used as the data is continuous 2C Functions h(x) Find the range of the following function, and state if it is one-toone or many-to-one. 6 Range h(x) = 1/x, domain {x є R, 0 < x ≤ 3} 8 4 2 h(x) = 1/x, {x є R, 0 < x ≤ 3} Inequality, so you will have to sketch the graph Range of h(x): h(x) ≥ 1/3 Description: One to One -3 -2 -1 1 2 h(x) = 1/x x 3 In this domain, the smallest value is 1/3 As we get close to 0, values will get infinitely high 2C Functions You will need to be able to plot more than one function on the same set of axes, possibly for different domains. f(x) = 5 – 2x 8 6 4 The function f(x) is defined by: f(x) = { 5 – 2x x<1 x2 + 3 x≥1 a) Sketch f(x) stating its range f(x) > 3 b) Find the values of a such that f(a) = 19 f(x) = x2 + 3 f(x) 2 -3 -2 -1 1 2 3 x Sketch both graphs on the same axes Make sure you use the correct domain for each The lowest value plotted is 3. Careful though as for 5 – 2x, x cannot include 1. Therefore f(x) > 3 (not including 3) 2C Functions You will need to be able to plot more than one function on the same set of axes, possibly for different domains. 8 6 The function f(x) is defined by: f(x) = { 5 – 2x 4 2 x<1 x2 + 3 x ≥ 1 a) Sketch f(x) stating its range f(x) > 3 b) Find the values of a such that f(a) = 19 Solve both equations separately! Remember that the answers must be within the domain given, or they cannot be included f(x) = x2 + 3 y f(x) = 5 – 2x -3 -2 -1 1 2 Linear Equation 5 – 2x = 19 Quadratic Equation x2 + 3 = 19 – 2x = 14 x2 = 16 x = -7 x = ±4 x=4 3 x (Has to be greater than 1) 2C Functions Combining Functions Two or more functions can be combined to make a more complex function. Given: f(x) = x2 g(x) = x + 1 Find: a) fg(x) b) gf(x) It helps to write what you would do to x for each function f(x) = x2 g(x) = x + 1 ‘Square x’ ‘Add 1 to x’ a) fg(x) means g acts first, followed by f. fg(x) f(x + 1) (x + 1)2 fg(x) = x2 + 2x + 1 Replace g(x) with the function f(x) means ‘square x’, so square g(x) Multiply out and simplify 2D Functions Combining Functions Two or more functions can be combined to make a more complex function. Given: f(x) = x2 g(x) = x + 1 Find: a) fg(x) = x2 + 2x + 1 b) gf(x) It helps to write what you would do to x for each function f(x) = x2 g(x) = x + 1 ‘Square x’ ‘Add 1 to x’ b) gf(x) means f acts first, followed by g. gf(x) g(x2) (x2) + 1 Replace f(x) with the function g(x) means ‘add one to x’, so add 1 to f(x) Simplify gf(x) = x2 + 1 2D Functions Combining Functions Two or more functions can be combined to make a more complex function. It helps to write what you would do to x for each function Given: f(x) = 3x + 2 g(x) = x2 + 4 Find: a) fg(x) b) gf(x) c) f2(x) d) The values of b so that fg(b) = 62 f(x) = 3x + 2 g(x) = x2 + 4 ‘Multiply by 3, then add 2’ ‘Square x then add 4’ a) fg(x) means g acts first, followed by f. fg(x) f(x2 + 4) 3(x2 + 4) + 2 fg(x) = 3x2 + 12 + 2 fg(x) = 3x2 + 14 Replace g(x) with the function f(x) means ‘multiply by 3, then add 2’ Multiply out and simplify 2D Functions Combining Functions Two or more functions can be combined to make a more complex function. It helps to write what you would do to x for each function Given: f(x) = 3x + 2 g(x) = x2 + 4 Find: a) fg(x) = 3x2 + 14 b) gf(x) c) f2(x) d) The values of b so that fg(b) = 62 f(x) = 3x + 2 g(x) = x2 + 4 ‘Multiply by 3, then add 2’ ‘Square x then add 4’ b) gf(x) means f acts first, followed by g. gf(x) g(3x + 2) (3x + 2)2 + 4 gf(x) = 9x2 + 12x + 4 + 4 gf(x) = 9x2 + 12x + 8 Replace f(x) with the function g(x) means ‘square then add 4’ Multiply out and simplify 2D Functions Combining Functions Two or more functions can be combined to make a more complex function. It helps to write what you would do to x for each function f(x) = 3x + 2 g(x) = x2 + 4 ‘Multiply by 3, then add 2’ ‘Square x then add 4’ c) f2(x) means f acts again on itself Given: f(x) = 3x + 2 g(x) = x2 + 4 Find: 3x2 a) fg(x) = + 14 b) gf(x) = 9x2 + 12x + 8 c) f2(x) d) The values of b so that fg(b) = 62 f2(x) f(3x + 2) 3(3x + 2) + 2 f2(x) = 9x + 6 + 2 f2(x) = 9x + 8 Replace f(x) with the function f(x) means ‘multiply by 3, then add 2’ Multiply out and simplify 2D Functions Combining Functions Two or more functions can be combined to make a more complex function. It helps to write what you would do to x for each function Given: f(x) = 3x + 2 g(x) = x2 + 4 Find: a) fg(x) = 3x2 + 14 b) gf(x) = 9x2 + 12x + 8 c) f2(x) = 9x + 8 d) The values of b so that fg(b) = 62 f(x) = 3x + 2 g(x) = x2 + 4 ‘Multiply by 3, then add 2’ ‘Square x then add 4’ d) fg(b) = 62, find b fg(b) = 62 3x2 + 14 = 62 3x2 = 48 x2 = 16 x=±4 Replace fg(b) with the function fg(x) Work through and solve the equation Remember 2 possible values 2D Functions Combining Functions Two or more functions can be combined to make a more complex function. Given: You need to work out what order of m, n and p will give the result when they are combined. The best way is to do some ‘trial and error’ mentally. Looking at what is in the equation helps speed this up. 1/ x 2/ x n(x) is the only function that contains doubling 1/ x m(x) = n(x) = 2x + 4 p(x) = x2 – 2 Find in terms of m, n and p, the function: a) 2/x + 4 m(x) has been doubled nm(x) n(1/x) 2(1/x) + 4 2/ + 4 x so we need m(x) to begin with, followed by n(x) Replace m(x) n(x) doubles and adds 4 Multiply out and simplify 2D Functions Combining Functions Two or more functions can be combined to make a more complex function. Given: m(x) = 1/x n(x) = 2x + 4 p(x) = x2 – 2 Find in terms of m, n and p, the function: b) 4x2 + 16x + 14 You need to work out what order of m, n and p will give the result when they are combined. The best way is to do some ‘trial and error’ mentally. Looking at what is in the equation helps speed this up. There is a x2 in the final answer, and no fraction, so most likely n and p are involved If we had the x2 part first, it would only get multiplied by 2, not 4. Whereas if we have ‘2x’ and square it, we get 4x2. Therefore n must come before p. pn(x) p(2x + 4) (2x + 4)2 - 2 4x2 + 16x + 14 Replace n(x) p(x) squares and subtracts 2 Multiply out and simplify 2D Functions Inverse Functions You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) Some simple inverses Function f(x) = x + 4 g(x) = 2x h(x) = 4x + 2 Inverse f-1(x) = x - 4 g-1(x) = x/2 h-1(x) = x – 2/4 2E Functions Inverse Functions Find the inverse of the following function You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) To calculate the inverse of a function, you need to make ‘x’ the subject f(x) = 3x2 - 4 y = 3x2 - 4 y + 4 = 3x2 y + 4/ 3 = x2 √(y + 4/3) = x The inverse is written ‘in terms of x’ +4 ÷3 Square root f-1(x) = √(x + 4/3) 2E Functions Find the inverse of the following function Inverse Functions You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) To calculate the inverse of a function, you need to make ‘x’ the subject m(x) = y = 3/ (x – 1) y(x – 1) = 3 yx - y = 3 yx = 3 + y The inverse is written ‘in terms of x’ x= 3 + y/ y 3/ (x – 1) Multiply by (x – 1) Multiply the bracket Add y Divide by y m-1(x) = 3 + x/x 2E Functions Inverse Functions You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) ‘The function f(x) is defined by f(x) = √(x – 2), x ε R, x ≥ 2’ ‘Find f-1(x), stating its domain. Sketch the graphs and describe the link between them. Finding f-1(x) f(x) = √(x – 2) y = √(x – 2) y2 = x - 2 Square Add 2 y2 + 2 = x f-1(x) = x2 + 2 The inverse is written ‘in terms of x’ 2E Functions Finding the domain of f-1(x) Inverse Functions ‘The domain and range of a function switch around for its inverse’ You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is 1(x) f- f(x) f(x) = √(x – 2) ‘The function f(x) is defined by f(x) = √(x – 2), x ε R, x ≥ 2’ x ‘Find f-1(x), stating its domain. Sketch the graphs and describe the link between them. f-1(x) = x2 + 2 Range for f(x) f(x) ≥ 0 Domain for f-1(x) x ≥ 0 2E Functions Sketching the graph of f-1(x) Inverse Functions Domain is x ≥ 0, so we can draw the graph for any values of x in this range You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is 1(x) f- f(x) f-1(x) = x2 + 2 ‘The function f(x) is defined by f(x) = √(x – 2), x ε R, x ≥ 2’ f-1(x), ‘Find stating its domain. Sketch the graphs and describe the link between them. f(x) = √(x – 2) x f-1(x) = x2 + 2, {x ε R, x ≥ 0} The link is that f(x) is reflected in the line y = x 2E Functions Inverse Functions You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) If g(x) is defined as: g(x) = 2x - 4 Domain x ≥ 0 g-1(x) = x + 4/ 2 Domain x ≥ -4 Range g(x) ≥ -4 Range g-1(x) ≥ 0 f(x) g(x) g-1(x) g(x) = 2x – 4, {x ε R, x ≥ 0}, x Calculate and sketch g(x) and g-1(x), stating the domain of g-1(x). Summary • We have learnt about functions • We have seen what is a function, and what isn’t • We have also learnt how to calculate more complex functions, as well as the inverse function