14 Functions Cambridge MathematicsInternationalIGCSE™
Peter Blythe, Emma Low, Andrew Manning, Karen Morrison, Raju Taniparti & S. M. Teo
Jasmine
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CAMBRIDGE IGCSE™ INTERNATIONAL MATHEMATICS: COURSEBOOK ii Contents How to use this book 00 How to use this series 00 Introduction 00 Using Graphic Display Calculators 00 Carrying out investigations 00 1 Number 00 2 Operations with numbers 00 3 Using number 00 4 Angles and bearings 00 5 Triangles, quadrilaterals and polygons 00 6 Indices 00 7 Introduction to algebra 00 8 Coordinate geometry 00 9 Simultaneous linear equations 00 10 Symmetry, congruency and similarity 00 11 Pythagoras’ theorem 00 12 Mensuration 00 13 Quadratic expressions 00 14 Functions 00 15 Trigonometry 00 16 Circle properties 00 17 Vectors and transformations 00 18 Functions 2 00 19 Sets 00 20 Descriptive statistics 00 21 Cumulative frequency graphs and linear regression 00 22 Probability 00 23 Sequences 00 24 Modelling 00 Answers 00 Glossary 00 Index 00 Acknowledgements 00 We are working with Cambridge Assessment International Education towards endorsement of this resource. Original material © Cambridge University Press 2022. This material is not final and is subject to further changes prior to publication. SAMPLE
How to use this book
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How to use this book
Exercises
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MATHEMATICS IN CONTEXT
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INVESTIGATION/DISCUSSION
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GETTING STARTED
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SUMMARY
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REFLECTION
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SELF/PEER ASSESSMENT
MATHEMATICAL CONNECTIONS
Past paper questions
These activities ask you to think about the approach that you take to your work, and how you might improve this in the future.
CAMBRIDGE IGCSE™ INTERNATIONAL MATHEMATICS: COURSEBOOK iv
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1 Chapter 14 Functions IN THIS CHAPTER YOU WILL: •use function notation • create tables for functions •sketch graphs of functions •use a graphic display calculator to explore functions •recognise function types from the shape of their graphs learn how to describe the range and domain of a function understand and find inverse functions form composite functions. SAMPLE We are working with Cambridge Assessment International Education towards endorsement of this resource. Original material © Cambridge University Press 2022. This material is not final and is subject to further changes prior to publication.
CAMBRIDGE IGCSE™ INTERNATIONAL MATHEMATICS: COURSEBOOK 2 GETTING STARTED What do you already know? 1 Calculate the value of y in each of these equations when x = 7 a y = 5x + 10 b y = 4x + 3 8 c y = 100 − x2 d y = x2 − 3x + 5 2 a Copy and complete this table for the equation y = 3x − 5 x 0123 y −5 (x, y)(0, −5) b Copy this grid and plot the coordinates from your table onto your grid. 1 1 −1−1Ox y −5−4−3−22345 2 3 4 5−2−3−5 −4 c Join your plotted points with a ruler. They should make a straight line. Label the line y = 3x − 5 3 a Draw and complete a table for the equation y = 2 − x b Draw a grid and plot the points you have recorded in your table for y = 2 − x c Draw a line through the points and label the line. CONNECTIONSMATHEMATICAL In Chapter 7 you learnt functions.areusefulThisintosubstitutingaboutvaluesequations.willbeverywhenyouworkingwith SAMPLE We are working with Cambridge Assessment International Education towards endorsement of this resource. Original material © Cambridge University Press 2022. This material is not final and is subject to further changes prior to publication.
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14 Functions 3
WORKED EXAMPLE 1 Consider the function f(x) = x2 + 2x − 1 a Describe what this function does in words. b Evaluate f(3). aAnswer f(x) = x2 + 2x − 1 The function squares the input (x), adds the input multiplied by two, then subtracts one. b Substitute x for 3 in the function. f(3) = 32 + (2 × 3) − 1 = 9 + 6 − 1 = 14 Exercise 14.1 1 Describe what each function does in words. a f(x) = 8x b f(x) = x 10 c f(x) = 3x − 10 d f(x) = x2 + 9 e f(x) = 2x2 − 4x + 3
realrangequadraticoutputlinearinverseinputfunctiondomainfunctionfunctionfunctionnumber
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A function relates an input to an output. The value that is entered into a function is called the input. The value that is returned by a function is called the output.
Algebraic equations are very useful for recording functions. However, using function notation allows you to do more with functions than the equations you have used Thisbefore.is a function written in function notation:f( x) = x2 − 5 f is the name of the function. You can use any letter. f is commonly used because it is the first letter of the word ‘function’. The value of f(x) is the output. This is what the function does to the input. x is the input.
14.1 Using function notation
KEY WORDS composite function
SAMPLE
4 2 Write each of these functions using function notation.
3 Which of these functions does not have an output of 10 when the input is x = −2?
TIP Don’t forget to give your answer in units based on the context of the problem. Think about what your answer represents.
Functions are a way of concisely describing many events that happen in real life. Functions can be used to calculate prices or predict outcomes.
Jo sells boat trips to groups of customers. Jo charges $21 for each customer and there is a booking fee for the group of $30.
a f(x) = −5x b g(x) = x + 12 c h(x) = x + 2 3 d f(x) = 8 + x e g(x) = 8 − x f h(x) = x2 + 10
The function squares the input (x), then divides by two. d The function squares the input (x), multiplies by eight, then subtracts the original input multiplied by five.
5 The mass of the biscuit box is 58 g. The mass of one biscuit is 13 g. a Express the total mass of the biscuit box and biscuits as a function where g(x) is the total mass and x is the number of biscuits. b Use your function to evaluate: i g(25) ii g(82) iii g(3.5)
TIP The notation f(10) is the value of f(x) when x = 10. So, this is the total cost of a boat trip for ten customers.
Question 4 asks you to use a function to calculate the cost of a boat trip. It is useful to use a function here because you can substitute x into the function to work out the total cost for any number of people.
MATHEMATICS
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IN CONTEXT
a The function multiplies the input (x) by 12, then subtracts four. b The function divides the input (x) by five, then adds seven. c
4
What do your answers to part b represent?
SAMPLE
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CAMBRIDGE IGCSE™ INTERNATIONAL MATHEMATICS: COURSEBOOK
REFLECTION
a Express the cost of a boat trip as a function where f(x) is the total cost and x is the number of customers. b Use the function to calculate: i f(10) ii f(57) iii f(129)
The input x 1234 The output f(x) 5 678 b 4321 8765 x f(x) f(1) = 1 + 4 = 5 f(2) = 2 + 4 = 6 f(3) = 3 + 4 = 7 f(4) = 4 + 4 = 8 This column shows that when x equals 1, f(x) equals 5. Calculate these outputs in the same way. 3 + 4 = 7, so y equals 7. 4 + 4 = 8, so y equals 8. The other row or column is for the outputs (f(x)) of the function for each input value. Calculate this output. The input is 2. 2 + 4 = 6. The output is 6. Write ‘6’ here. One row or column is for possible input (x)values.
Tables and mapping diagrams
14 Functions 5 14.2
WORKED EXAMPLE 2 For the function f(x) = x + 4, copy and complete: a the table of values The input x 1234 The output f(x) 5 b the mapping diagram. 4321 ……5 x f(x) aAnswer
SAMPLE
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You can record the inputs and outputs for a function in a table or with a mapping diagram. You can then use these sets of values to explore and understand what a function does.
CAMBRIDGE IGCSE™ INTERNATIONAL MATHEMATICS: COURSEBOOK Exercise6 14.2 1 Copy and complete the table of values for the function f(x) = 2x + 3 Input (x)Output (f(x)) 1 5 2 7 6543 2 This table shows some inputs and outputs of a function. Input 12345 Output 3691215 a Describe the function in words. b Write the function using function notation. 3 Copy and complete the mapping diagram of values for the function g(x) = 5x − 1 −1−2−3012 −16…9 x g(x) 4 Copy and complete the equation to describe the function h(x). x 123456 h(x) 112131415161 h(x) = x + 5 Copy and complete the mapping diagram of values for the function f(x) = 12 x 654321 2.412… x f(x) TIP The input is 1. 1 multiplied by 2 equals 2. 2 add 3 equals 5. The output is 5. TIP x = −3 −3 × 5 = −15 −15 − 1 = −16 g(−3) = −16 TIP It is not possible to divide a number by 0, so the input for this function cannot be 0. SAMPLE We are working with Cambridge Assessment International Education towards endorsement of this resource. Original material © Cambridge University Press 2022. This material is not final and is subject to further changes prior to publication.
TIP Choose consecutive integers for your input values (x). Include x = 0. TIP x2 means −(x2), not (−x)2. Square x first, then make it negative.
INVESTIGATION 1 Investigate different integer inputs and outputs for the function f(x) = 2x − 1. Record your inputs and outputs in a table. Describe what property the outputs for this function have in common. Find a function where the output will never be a negative number. Explain why your function will never produce a negative number.
e.
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14 Functions 7 6 Copy and complete the table of values for the function g(x) = x2 − 1 x −3−2−1012 g(x) 7 Copy and complete the table of values for the function f(x) = x2 + 4x x −2−10123 x2 4 4x −8 f(x) −4 8 Copy and complete the table of values for the function f(x) = x2 + 7x + 10 x x 72x +10 +10+10+10+10+10+10 f(x) 9 Draw and complete your own tables of values for these functions. a f(x) = x2 + x − 1 b g(x) = x2 + 4x − 7 c h(x) = −x2 + 2x + 2 d f(x) = −x2 −x + 5
TIP In this table you calculate the two parts of the function in separate rows. You then add the two parts to calculate the output values for f(x).
CAMBRIDGE IGCSE™ INTERNATIONAL MATHEMATICS: COURSEBOOK 14.38 Graphs You can represent a function on a graph which helps you to see, understand and use the relationship between inputs and outputs. The input is along the horizontal axis and the output is along the vertical axis. WORKED EXAMPLE 3 Sketch the graph of the function f(x) = 2x − 3 Answer First create a table of values for the function. x −10123 f(x) −5−3−113 Draw a pair of axes. Make sure that all your x values fit on the horizontal axis and all of your f(x) values fit on the vertical Convertaxis.the values in your table into coordinates (x, f(x)). Then plot the coordinates onto your grid. x −10123 f(x) −5-3−113 (x, f(x)) (−1, −5)(0, −3)(1, −1)(2, 1)(3, 3) 1 1 −1−1Ox f(x −5−4−3−2)2345 2 3 4 5−2−3−5 −4 1 1 −1−1Ox f(x −5−4−3−2)2345 2 3 4 5−2−3−5 −4 SAMPLE We are working with Cambridge Assessment International Education towards endorsement of this resource. Original material © Cambridge University Press 2022. This material is not final and is subject to further changes prior to publication.
14 Functions 9
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CONTINUED Draw a line through the plotted points. A linear function, in the form f(x) = ax + b creates a straight line graph. If the points are in a straight line, use a ruler to join them. Label the line with the function. 1 1 −1−1Ox f(x −5−4−3−2)2345 2 3 4 5−2−3−5 −4 f(x) = 2x − 3 Exercise 14.3
SAMPLE
2 a Draw and complete a mapping diagram for the function f(x) = 2 − x b Convert the inputs and outputs recorded in your mapping diagram to coordinates (x, f(x)). Draw a grid and plot the points from your mapping diagram. c Draw a line through the points and label it with the function.
1 a Draw and complete a table of values for the function f(x) = x − 2 b Draw a grid and plot the points you have recorded in your table for y = f(x). c Draw a line through the points and label it with the function.
3 a Copy and complete the table for the function f(x) = x2 + 2x − 4 x −2−1012 x2 41 2x −4−2 −4 −4−4 f(x) −4 b Sketch the graph of f(x).
TIP The x 2 in question 3 makes this a quadratic function. Quadratic functions have the general form f(x) = ax 2 + bx + c A graph of a quadratic function will look similar to 1 1 −1−1Ox y −3−223 2 3−2−3 when a is positive or 1 1 −1−1Ox y −3−223 2 3−2−3 when a is negative. Join the points on the graph with a smooth curve, not a straight line.
Exercise 14.4
Use your calculator to find the coordinates of: a the local maximum where −2 < x < 0 b the local minimum where 0 < x < 2 c the zeros of the graph of f(x).
a
Graphical display calculators and recognising function types by the shape of their graph
You can enter a function into your graphical display calculator. You can then use the function mode on your calculator to explore features of the graph of the function.
Give your answers correct to three significant figures where appropriate.
2 Select the table function mode on your graphical display calculator. Enter the function f(x) = 7x2 − 3
Use your calculator to find the intersections of the functions. Round solutions to two decimal places.
Use your calculator to create a table of values from x = −2 to x = 1, in intervals of Copy0.5.the table.
TIP You will need to identify how to change settings to specify the values of x in the table. Check with your teacher or the instructionscalculatorifneeded.
3 Sketch both these functions on your graphical display calculator. f(x) = −2x2 +3x − 1 g(x) = −x − 2
TIP A quadratic function has the general form f(x) = ax2 + bx + c If a in the quadratic function is negative, the graph will have a maximum value vertex. If a in the quadratic function is positive, the graph will have a minimum value vertex.
TIP A local maximum on a graph is the point on the line that has the highest value of y in a given part of the graph. A local minimum on a graph is the point on the line that has the lowest value of y in a given part of the graph.
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On your graphical display calculator sketch the graph of the function f(x) = −x2 + 2x + 1 i Find the vertex of the graph of the function. ii Find the zeros of the graph of the function.
CAMBRIDGE IGCSE™ INTERNATIONAL MATHEMATICS: COURSEBOOK 14.410
1 Select the graph function mode on your graphical display calculator. Use the graph menu to reveal features of these graphs.
4 On your graphical display calculator sketch the graph of the function f(x) = x3 − 3x + 0.5
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b On your graphical display calculator sketch the graph of the function f(x) = x2 − 2x − 6 i Find the vertex of the graph of the function. ii Find the zeros of the graph of the function.
SAMPLE
14 Functions 11
1 Linear. In the form f(x) = ax + b, for example f(x) = 3x − 2 2 Quadratic. In the form f(x) = ax 2 + bx + c, for example f(x) = x 2 + 2x 3 Cubic. In the form f(x) = ax 3 + bx 2 + cx + d, for example f(x) = x 3 + 5 4 Reciprocal. In the for m a x , for example f(x) = 1 x Select the graph function mode on your graphical display calculator. Enter at least three different linear functions and look at the graphs they create. Describe the shape of the graph made by the functions. How are the graphs Entersimilar?at least three different quadratic functions and look at the graphs they Describecreate. the shape of the graph made by the functions. How are the graphs Repeatsimilar? with cubic and reciprocal functions.
PEER ASSESSMENT
SAMPLE
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Discuss with your partner, or in a small group, what you have learnt about using your graphical display calculator to explore functions. Quiz each other on how to: • sketch a graph • produce a table of values find the vertex of a quadratic function (maximum or minimum) find the coordinates of an intersection of two graphs of functions • find the zeros, local maximum or minimum.
INVESTIGATION 2 Here are four types of function.
REFLECTION Reflect on your choices of function in Investigation 2. Did you choose a variety of different functions including: • functions with negative and positive values • functions with integer and decimal values • functions with very high or very low values? Discuss with a partner or in a small group why it might be important to choose a variety of different functions when investigating the properties of the graphs of these functions.
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CAMBRIDGE IGCSE™ INTERNATIONAL MATHEMATICS: COURSEBOOK 12 5 a Which of these graphs represent a linear function? b Which of these graphs represent a quadratic function? A 1 −1−1Ox y −5−4−3−22 132 4−2−3−4 B 1 1 −1−1Ox y −2234 2 3 4−2−3−4 C 1 1 −1−1Ox y −2234 2 3 4−2−3−4 D 1 1 −1−1Ox y 5432 2 3 4 5 6 7−2 E 1 1 −1−1Ox y −2234 2 3 4−2−3−4 F 1 −1−1Ox y −4−3−22 132 4−2−3−4 G 1 1 −1−1Ox y −3−223 2 3 4−2−3−4 H 1 1 −1−1Ox y −2234 2 3 4−2−3−4 6 Look again at the graphs in Question 5. a Which graphs represent a cubic function? b Which graphs represent a reciprocal function? CONNECTIONSMATHEMATICAL You will be able to apply what you have learnt about equations in coordinate geometry in Chapter 8 to the graphs of functions. SAMPLE We are working with Cambridge Assessment International Education towards endorsement of this resource. Original material © Cambridge University Press 2022. This material is not final and is subject to further changes prior to publication.
14 Functions 13 7 Enter each of these functions into your graphical display calculator.
a f(x) = 3x + 1 b f(x) = −2x − 2 c f(x) = −x + 1 d f(x) = 2x − 3 e f(x) = −3x + 2 f f(x) = 2x + 2 A 1 1 −1 −1 Ox y −3−223 2 3−2−3 B 1 1 −1 −1 Ox y −3−223 2 3−2−3 C 1 1 −1 −1 Ox y −3−223 2 3−2−3 D 1 1 −1 −1 Ox y −3−223 2 3−2−3 E 1 1 −1 −1 Ox y −3−223 2 3−2−3 F 1 1 −1 −1 Ox y −3−223 2 3−2−3 TIP If necessary, change the minimum and maximum values of the horizontal and vertical axes so that you can see the place where the function line intercepts with the vertical axis.
SAMPLE
FunctionVertical axis interceptGradient f(x) = x + 1 f(x) = 2x + 1 f(x) = 3x + 2 f(x) = 3x − 1 f(x) = −2x + 3 f(x) = −3x − 2 b For a linear graph in the form f(x) = ax + b, describe the link between the function and where the graph of the function intercepts the vertical axis.
8 Use what you know about graphs of functions, the vertical axis intercept and the gradient of a line to match each function to its graph.
a Copy and complete the table to show the features of each graph.
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c For a linear graph in the form f(x) = ax + b, describe the link between the function and the gradient of the graph of the function.
SAMPLE
WORKED EXAMPLE 4 a What is the domain of the function f(x) = √ x + 1 ? b What is the range of the function f(x) = √ x + 1 ? aAnswer
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Think about which values of x produce a real number output f(x).
The domain refers to the set of possible real number inputs that will produce a real number output when the function is applied. Many functions can have an input that is any number from −∞ to ∞ and it will create an output. However, for some functions there are some input values that do not create an output value. The range of the function describes the set of all the possible output values for the function. If you enter the function into a graphical display calculator it should give you an idea as to which values are possible output values of the function.
The square root of a negative number is not a real number, so x + 1 cannot be negative. If x is −2, then x + 1 = −1. √ −1 does not produce a real number output. x cannot equal −2, so −2 is not in the domain of f(x). Any value for x less than −1 is not in the domain of f(x). The domain of f(x) is x > −1 1 1 −1 −1 Ox y 32 2 3−2 b Enter the function into your graphical display calculator. This is the graph of f(x). The vertical axis represents the output of the function. There are no values of y less than 0 because x must be greater than or equal to −1. The range of f(x) is f(x) > 0
CAMBRIDGE IGCSE™ INTERNATIONAL MATHEMATICS: COURSEBOOK 14.514
The domain and the range
a f(x) = √ x The domain of f(x) is ... b g(x) = √ 5 − x The domain of g(x) is ... c h(x) = 7 √ x − 3 The domain of h(x) is ... 3 What is the range of each function? a f(x) = x2 b g(x) = 1 x c h(x) = x + 6 3 when 0 < x < 9 d k(x) = 3x2 − 10 when −5 < x < 5
14.6
TIP For a value to be a possible input value, the output created must be a real number. For part a, use your knowledge of number and operations to consider what number five cannot be divided by. TIP Domains are specified for functions h(x) and k(x). A domain specified in this way is called a restricted domain. Calculate the possible outputs for the specified domain only.
SAMPLE
Here is the function f(x) = 5x − 8 represented as a function machine: input (x) ×5 −8 output (f(x)) To calculate f(10) you multiply 10 by five and then subtract eight. f(10) = 42
Inverse functions
1 a For each of these functions there is a value, or values, of x that is impossible. Find the impossible value(s) of x i f(x) = 5 x ii g(x) = 2x + 2 x − 8 iii h(x) = 7 xx 2 − 36 b Explain why the values you identified in part a are impossible input values for the functions. 2 For each of these functions there is a set of input values that are impossible. Copy and complete the statements to describe the set of values for x that will give a real number defined output for each function.
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14 Functions 15 Exercise 14.5
An inverse function takes an output of the function and calculates what the input was. You use the notation f–1(x) to describe the inverse function for f(x). WORKED EXAMPLE 5 Consider the function f(x) = 5x − 8 a Find f −1(x). b If f(x) = 27, what is the input value x?
Let y = f(x Rearrange) to make x the subject: y = 5x − 8 y + 8 = 5x y + 8 5 = x Swap x for f −1(x), and swap y for x: x = y + 8 5 f −1(x) = x + 8 5 b Substitute x = 27 into f −1(x) to find the value of x that gives f(x) = 27: f −1(27) = 27 + 8 5 = 7 x = 7 Exercise 14.6 1 a For the function f(x) = 3x − 4, express the inverse function in the form f −1(x) = ... b For the function g(x) = x + 2 3 , express the inverse function in the form g−1(x) = ... c For the function h(x) = x2 + 8, express the inverse function in the form h−1(x) = ... d For the function k(x) = 5 − x, express the inverse function in the form k−1(x) = ...
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16CONTINUED
aAnswerMethod
1: Using a function machine input (x) ×5 −8 output (27) Reverse multiplying by 5 by dividing by 5. 35 ÷ 5 = 7 x = 7 Reverse subtracting 8 by adding 8. 27 + 8 = 35
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Rename the output as x and write this inverse function as: f −1(x) = x + 8 5 Method 2: Using changing the subject in an equation You can also find the inverse function by changing the subject in the original function.
SAMPLE
• What are the benefits and disadvantages of each method?
2 f(x) = 7x − 10, g(x) = 8 − 2x, h(x) = 3 x +1 Consider your answers to the reflection questions as you find: a f −1(x) b g−1(x) c h−1(x).
• Are there any questions that can only be solved using one of the methods?
14.7 Composite functions Two functions can be combined into a single function called a composite function. In a composite function, one function is the input (x) of the other function. fg(x) g(x) is a function. In fg(x), g(x) is the input of function f. This means you need to carry out the function g(x) first, then apply function f(x) to the output of g(x). Always apply the function closest to the x is first.
14 Functions 17 REFLECTION Look back at Worked example 5 to recap the two methods for finding the inverse of a function.
WORKED EXAMPLE 6 f(x) = 5x − 3 and g(x) = x 2 Find: a i fg(x) ii fg(4) b i gf(x) ii gf(4). aAnsweri fg(x) = f(g(x)) = f( x 2 ) = 5( x 2 ) − 3 = 5x 2 − 3 ii When x = 4 fg(4) = 5 × 4 2 −3 = 7 b i gf(x) = g(f(x)) = g(5x − 3) = 5x − 3 2 ii When x = 4 gf(4) = 5 × 4 − 3 2 = 8.5 TIP Usually fg(x) ≠ gf(x) g(x) is the input. f(x) is the input. We are working with Cambridge Assessment International Education towards endorsement of this resource.
SAMPLE
• Which method(s) did you use for the questions in this exercise? Why?
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CAMBRIDGE IGCSE™ INTERNATIONAL MATHEMATICS: COURSEBOOK Exercise18 14.7 1 f(x) = 2x + 1 and g(x) = x2 − 3 Find: a i fg(x) ii fg(5) b i gf(x) ii gf(5). 2 f(x) = x 4 , g(x) = 3x2 and h(x) = −2x + 5 Find: a fg(8) b gh(3) c f h(2) d gf(−10) e hg(0.8) f hf(p) (round your answer to two decimal places). 3 f(x) = 6 x + 1 , g(x) = (2x − 3)2 and h(x) = x − 3 8 Find: a gf(2) b fg(5) c gh(3) d hg( 12 ) e f h(67) f hf(−7). SUMMARYAreyouable to... ? use function notation create tables for functions sketch graphs of functions use a graphic display calculator to explore functions recognise function types from the shape of their graphs Do you know how to... ? describe the range and domain of a function understand and find inverse functions form composite functions TIP If your answer is a fraction, give your answer in its simplest form. SAMPLE We are working with Cambridge Assessment International Education towards endorsement of this resource. Original material © Cambridge University Press 2022. This material is not final and is subject to further changes prior to publication.
1 f(x) = (2x − 3)(x − 1) Work out f(7). [2] [CambridgeIGCSEInternationalMathematics0607Paper12Q10May/June2021]
Past paper questions
3 f(x) = x − 3 2 for −5 < x < 21 Find the range of f(x). [2] [CambridgeIGCSEInternationalMathematics0607Paper11Q20May/June2021]
SAMPLE
We are working with Cambridge Assessment International Education towards endorsement of this resource. Original material © Cambridge University Press 2022. This material is not final and is subject to further changes prior to publication.
The domain of f(x) is 1 < x < 9. Find the range of f(x). [2] [CambridgeIGCSEInternationalMathematics0607Paper13Q23May/June2021]
14 Functions 19
2 f(x) = x − 3
4 f(x) = 2x − 1 g(x) = 3 − x h(x) = x2 a Find i f(−2), [1] ii h(g(−2)). [2] b Solve f(x) = 7. x = [2] c Find f(g(x)). [1] d Solve f(x) × g(x) + 2h(x) = 0. x = [3] e Find g−1(x). g−1(x) = [2] [CambridgeIGCSEInternationalMathematics0607Paper41Q5May/June2021]