Mappings and Functions, Summaries of Algebra

Download Mappings and Functions and more Summaries Algebra in PDF only on Docsity! 4 Functions Still glides the stream and shall forever glide; The form remains, the function never dies. William Wordsworth Why fly to Geneva in January? Several people arriving at Geneva airport from London were asked the main purpose of their visit. Their answers were recorded. David Joanne Skiing Jonathan Returning home Louise To study abroad Paul This is an example of a mapping. Shamaila Karen Business 106 The language of functions A mapping is any rule which associates two sets of items. In this example, each of the names on the left is an object, or input, and each of the reasons on the right is an image, or output. For a mapping to make sense or to have any practical application, the inputs and outputs must each form a natural collection or set. The set of possible inputs (in this case, all of the people who flew to Geneva from London in January) is called the domain of the mapping. The seven people questioned in this example gave a set of four reasons, or outputs. These form the range of the mapping for this particular set of inputs. Notice that Jonathan, Louise and Karen are all visiting Geneva on business: each person gave only one reason for the trip, but the same reason was given by several people. This mapping is said to be many-to-one. A mapping can also be one-to- one, one-to-many or many-to-many. The relationship between the people from any country and their passport numbers will be one-to-one. The relationship between the people and their items of luggage is likely to be one-to-many, and that between the people and the countries they have visited in the last 10 years will be many-to-many. Mappings In mathematics, many (but not all) mappings can be expressed using algebra. Here are some examples of mathematical mappings. 4 (a) Domain: integers Range Objects Images 1 3 0 5 1 7 2 9 3 11 General rule: x 2x  5 (b) Domain: integers Range Objects Images 1.9 2 2.1 2.33 2.52 3 2.99 π General rule: Rounded whole numbers Unrounded numbers (c) Domain: real numbers Range Objects Images 0 45 0 90 0.707 135 1 180 General rule: x° sin x° (d) Domain: quadratic Range equations with real roots Objects Images x2  4x  3  0 0 x2  x  0 1 x2  3x  2  0 2 3 General rule: ax2  bx  c  0 x  –b – x  –b  b2 – 4ac 2a b2 – 4ac 2a 107 T h e la n g u a g e o f fu n c tio n s P 1 4 P1 y y = 2x + 1 O x y y = x3 – x –1 O 1 x y 1 y = ±2x O x –1 y 5 y = ± 25 – x2 –5 O 5 x domain: –5  x  5 –5 Figure 4.3 illustrates some different types of mapping. The graphs in (a) and (b) illustrate functions, those in (c) and (d) do not. (a) One-to-one (b) Many-to-one (c) One-to-many (d) Many-to-many Figure 4.3 1 Describe each of the following mappings as either one-to-one, many-to- one, one-to-many or many-to-many, and say whether it represents a function. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) 110 EXERCISE 4A F u n c ti o n s  x x  3 2 For each of the following mappings: (a) write down a few examples of inputs and corresponding outputs (b) state the type of mapping (one-to-one, many-to-one, etc.) 4 (c) suggest a suitable domain. (i) Words  number of letters they contain (ii) Side of a square in cm  its perimeter in cm (iii) Natural numbers  the number of factors (including 1 and the number itself) (iv) x  2x  5 (v) x  (vi) The volume of a sphere in cm3  its radius in cm (vii) The volume of a cylinder in cm3  its height in cm (viii) The length of a side of a regular hexagon in cm  its area in cm2 (ix) x  x2 3 (i) A function is defined by f(x)  2x  5, x  . Write down the values of (a) f(0) (b) f(7) (c) f(3). (ii) A function is defined by g:(polygons)  (number of sides). What are (a) g(triangle) (b) g(pentagon) (c) g(decagon)? (iii) The function t maps Celsius temperatures on to Fahrenheit temperatures. It is defined by t: C  9C  32, C  . Find5 (a) t(0) (b) t(28) (c) t(10) (d) the value of C when t(C)  C. 4 Find the range of each of the following functions. (You may find it helpful to draw the graph first.) 1 (ix) f(x) 1  x 2 (x) f(x)  x    3 x  3 5 The mapping f is defined by f(x)  x2 0  x  3 f(x)  3x 3  x  10. The mapping g is defined by g(x)  x2 0  x  2 g(x)  3x 2  x  10. Explain why f is a function and g is not. 111 E x e rc is e 4 A P 1 ( i ) f(x)  2  3x x  0 ( i i ) f(θ)  sin θ 0°  θ  180° ( ii i ) y  x2  2 x  {0, 1, 2, 3, 4} ( y  tan θ 0°  θ  90° 5 P1 Composite functions It is possible to combine functions in several different ways, and you have already 4 met some of these. For example, if f(x)  x 2 and g(x)  2x, then you could write f(x)  g(x)  x 2  2x. In this example, two functions are added. Similarly if f(x)  x and g(x)  sin x, then f(x).g(x)  x sin x. In this example, two functions are multiplied. Sometimes you need to apply one function and then apply another to the answer. You are then creating a composite function or a function of a function. A new mother is bathing her baby for the first time. She takes the temperature of the bath water with a thermometer which reads in Celsius, but then has to convert the temperature to degrees Fahrenheit to apply the rule that her own mother taught her: At one o five He’ll cook alive But ninety four is rather raw. Write down the two functions that are involved, and apply them to readings of (i) 30°C (ii) 38°C (iii) 45°C. SOLUTION The first function converts the Celsius temperature C into a Fahrenheit temperature, F. F  9C  32 The second function maps Fahrenheit temperatures on to the state of the bath. F  94 too cold 94  F  105 all right F  105 too hot This gives (i) 30°C  86°F  too cold (ii) 38°C  100.4°F  all right (iii) 45°C  113°C  too hot. 112 EXAMPLE 4.2 F u n c ti o n s This is a short way of writing x is an integer. Inverse functions P1 Look at the mapping x  x  2 with domain the set of integers. Domain Range 4 … … … … 1 1 0 0 1 1 2 2 … 3 … 4 x x  2 The mapping is clearly a function, since for every input there is one and only one output, the number that is two greater than that input. This mapping can also be seen in reverse. In that case, each number maps on to the number two less than itself: x  x  2. The reverse mapping is also a function because for any input there is one and only one output. The reverse mapping is called the inverse function, f1. Function: f : x  x  2 x  . Inverse function: f1 : x  x  2 x  . For a mapping to be a function which also has an inverse function, every object in the domain must have one and only one image in the range, and vice versa. This can only be the case if the mapping is one-to-one. So the condition for a function f to have an inverse function is that, over the given domain, f represents a one-to-one mapping. This is a common situation, and many inverse functions are self-evident as in the following examples, for all of which the domain is the real numbers. f : x  x  1; f1 : x  x  1 g : x  2x; g1 : x  1 x 2 h : x  x 3; h1 : x  3 x ●? Some of the following mappings are functions which have inverse functions, and others are not. (a) Decide which mappings fall into each category, and for those which do not have inverse functions, explain why. (b) For those which have inverse functions, how can the functions and their inverses be written down algebraically? 115 In v e rs e fu n c tio n s 4 P1 y y = f(x) O –1 1 x y y = g(x) O x y y = h(x) O x f(x) f(x) = x2 4 –2 O 2 x x (i) Temperature measured in Celsius  temperature measured in Fahrenheit. (ii) Marks in an examination  grade awarded. (iii) Distance measured in light years  distance measured in metres. (iv) Number of stops travelled on the London Underground  fare. You can decide whether an algebraic mapping is a function, and whether it has an inverse function, by looking at its graph. The curve or line representing a one- to-one function does not double back on itself and has no turning points. The x values cover the full domain and the y values give the range. Figure 4.5 illustrates the functions f, g and h given on the previous page. Figure 4.5 Now look at f(x)  x2 for x   (figure 4.6). You can see that there are two distinct input values giving the same output: for example f(2)  f(2)  4. When you want to reverse the effect of the function, you have a mapping which for a single input of 4 gives two outputs, 2 and 2. Such a mapping is not a function. Figure 4.6 You can make a new function, g(x)  x2 by restricting the domain to  (the set of positive real numbers). This is shown in figure 4.7. The function g(x) is a 116 one-to-one function and its inverse is given by g1(x)  means ‘the positive square root of’. since the sign F u n c ti o n s Single output value g(x) = x2, x  + Single input value P1 4 O x Figure 4.7 It is often helpful to define a function with a restricted domain so that its inverse is also a function. When you use the inv sin (i.e. sin–1 or arcsin) key on your calculator the answer is restricted to the range –90° to 90°, and is described as the principal value. Although there are infinitely many roots of the equation sin x = 0.5 (…, –330°, –210°, 30°, 150°, …), only one of these, 30°, lies in the restricted range and this is the value your calculator will give you. The graph of a function and its inverse ACTIVITY 4.1 For each of the following functions, work out the inverse function, and draw the graphs of both the original and the inverse on the same axes, using the same scale on both axes. (i) f(x)  x2, x  (ii) f(x)  2x, x  (iii) f(x)  x  2, x  (iv) f(x)  x3  2, x  Look at your graphs and see if there is any pattern emerging. Try out a few more functions of your own to check your ideas. Make a conjecture about the relationship between the graph of a function and its inverse. You have probably realised by now that the graph of the inverse function is the same shape as that of the function, but reflected in the line y  x. To see why this is so, think of a function f(x) mapping a on to b; (a, b) is clearly a point on the graph of f(x). The inverse function f1(x), maps b on to a and so (b, a) is a point on the graph of f1(x). The point (b, a) is the reflection of the point (a, b) in the line y  x. This is shown for a number of points in figure 4.8. 117 In v e rs e fu n c tio n s y    x y = f(x) y = x y = f–1(x) 4 P1 x  2 51  2 x x  4 x  8 x  4 The full definition of the inverse function y is therefore: f1(x)  for x  2. The function and its inverse function are shown in figure 4.11. (ii) f(7)  72  2  51 f1f(7)  f 1(51)   7 O x Figure 4.11 Applying the function followed by its inverse brings you back to the original input value. Note Part (ii) of Example 4.6 illustrates an important general result. For any function f(x) with an inverse f1(x), f1f(x)  x. Similarly ff1(x)  x. The effects of a function and its inverse can be thought of as cancelling each other out. 1 The functions f, g and h are defined for x  by f(x)  x3, g(x)  2x and h(x)  x  2. Find each of the following, in terms of x. (i) fg (ii) gf (iii) fh (iv) hf (v) fgh (vi) ghf (vii) g2 (viii) (fh)2 (ix) h2 2 Find the inverses of the following functions. (i) f(x)  2x  7, x  (ii) f(x)  4  x, x  (iii) f(x) 4 , x  2 (iv) f(x) x2 3, x  0 2 – x 3 The function f is defined by f(x)  (x  2)2  3 for x  2. (i) Sketch the graph of f(x). (ii) On the same axes, sketch the graph of f1(x) without finding its equation. 4 Express the following in terms of the functions f: x  x  0. and g: x  x  4 for (i) x  (iii) x  5 A function f is defined by: f: x  1 x  , x  0. (ii) x  x  8 (iv) x  Find (i) f 2(x) (ii) f 3(x) (iii) f1(x) (iv) f 999(x). 120 EXERCISE 4B F u n c ti o n s 6 (i) Show that x2  4x  7  (x  2)2  a, where a is to be determined. (ii) Sketch the graph of y  x2  4x  7, giving the equation of its axis of symmetry and the co-ordinates of its vertex. 4 The function f is defined by f: x  x2  4x  7 with domain the set of all real numbers. (iii) Find the range of f. (iv) Explain, with reference to your sketch, why f has no inverse with its given domain. Suggest a domain for f for which it has an inverse. 7 The function f is defined by f : x  4x3  3, x  . Give the corresponding definition of f1. State the relationship between the graphs of f and f1. [MEI] [UCLES] 8 Two functions are defined for x   as f(x)  x 2 and g(x)  x 2  4x  1. (i) Find a and b so that g(x)  f(x  a)  b. (ii) Show how the graph of y  g(x) is related to the graph of y  f(x) and sketch the graph of y  g(x). (iii) State the range of the function g(x). (iv) State the least value of c so that g(x) is one-to-one for x  c. (v) With this restriction, sketch g(x) and g1(x) on the same axes. 9 The functions f and g are defined for x   by f : x  4x  2x2; g : x  5x  3. (i) Find the range of f. (ii) Find the value of the constant k for which the equation gf(x) = k has equal roots. [Cambridge AS & A Level Mathematics 9709, Paper 12 Q3 June 2010] 10 Functions f and g are defined by f : x  k – x for x , where k is a constant, g : x  9 x  2 for x , x  –2. (i) Find the values of k for which the equation f(x) = g(x) has two equal roots and solve the equation f(x) = g(x) in these cases. (ii) Solve the equation fg(x) = 5 when k = 6. (iii) Express g–1(x) in terms of x. [Cambridge AS & A Level Mathematics 9709, Paper 1 Q11 June 2006] 121 E x e rc is e 4 B P 1 11 The function f is defined by f : x  2x2 – 8x + 11 for x . (i) Express f(x) in the form a(x + b)2 + c, where a, b and c are constants. 4 (ii) State the range of f. (iii) Explain why f does not have an inverse. The function g is defined by g : x  2x2 – 8x + 11 for x  A, where A is a constant. (iv) State the largest value of A for which g has an inverse. (v) When A has this value, obtain an expression, in terms of x, for g–1(x) and state the range of g–1 [Cambridge AS & A Level Mathematics 9709, Paper 1 Q11 November 2007] 12 The function f is defined by f : x  3x – 2 for x . (i) Sketch, in a single diagram, the graphs of y = f(x) and y = f– 1(x), making clear the relationship between the two graphs. The function g is defined by g : x  6x – x2 for x . (ii) Express gf(x) in terms of x, and hence show that the maximum value of gf(x) is 9. The function h is defined by h : x  6x – x2 for x  3. (iii) Express 6x – x2 in the form a – (x – b)2, where a and b are positive constants. (iv) Express h–1(x) in terms of x. [Cambridge AS & A Level Mathematics 9709, Paper 1 Q10 November 2008] 122 KEY POINTS 1 A mapping is any rule connecting input values (objects) and output values (images). It can be many-to-one, one-to-many, one-to-one or many-to-many. 2 A many-to-one or one-to-one mapping is called a function. It is a mapping for which each input value gives exactly one output value. 3 The domain of a mapping or function is the set of possible input values (values of x). 4 The range of a mapping or function is the set of output values. 5 A composite function is obtained when one function (say g) is applied after another (say f). The notation used is g[f(x)] or gf(x). 6 For any one-to-one function f(x), there is an inverse function f−1(x). 7 The curves of a function and its inverse are reflections of each other in the line y = x. F u n c ti o n s P 1