Cambridge IGCSE™ and O Level Additional Mathematics
EXAM PREPARATION AND PRACTICE
Nathan Barker, Martin Crozier & Lee Mckelvey with Digital access
1 Functions
KNOWLEDGE FOCUS
In this chapter, you will answer questions on:
• understanding functions, their domains and their ranges
• forming and using composite functions
• sketching modulus functions y = |f(x)|, where f(x) is linear, and solving equations involving the modulus function
• knowing when a function has an inverse and finding the inverse
• knowing the graphical relationship between a one–one function and its inverse.
EXAM SKILLS FOCUS
In this chapter you will:
• show that you understand the ‘state’ command word and can answer a ‘state’ question
• show that you understand the ‘sketch’ command word and can answer a ‘sketch’ question.
A very important part of understanding examination questions is recognising command words and knowing what they mean in a mathematics examination. In this chapter you will focus on two command words: ‘state’ and ‘sketch’.
State express in clear terms.
Sketch make a simple freehand drawing showing the key features.
Questions which use the command word ‘state’ are often looking for your understanding of the key concept in the topic and you do not usually need to write too much. You can give an answer without mathematical reasoning and ‘state’ questions do not usually require any calculations.
SAMPLE
The command word ‘sketch’ means that you must produce a simple drawing. You do not need to use graph paper or work out a set of coordinate points to produce an accurately plotted graph. However, you should always draw clearly labelled axes and try to show the shape and approximate position of the line or curve as accurately as you can. You should label the most important features, such as any intercepts, maximum or minimum points, or asymptotes. If a domain has been specified, make sure that your sketch covers all the domain and does not extend beyond it. If no domain is specified, your sketch should show any long-term behaviour, for example if the graph tends towards one of the coordinate axes.
2 For each pair of functions, find fg(x) and gf(x).
a f(x) = x2 − 2 for x ∈ ℝ g(x) = 5 x for x ∈ ℝ, x ≠ 0
b f(x) = 1 √ 3 + x for x ∈ ℝ, x > − 3 g(x) = 2 x 2 for x ∈ ℝ, x > 0
3 f(x) = 1 x − 2 for x ∈ ℝ, x ≠ 2 g(x) = 3x + 1 for x ∈ ℝ
a Solve fg(x) = 1 2 .
b Find gf(x), giving your answer in the for m ax + b cx + d
c Find ff(x), giving your answer in the for m ax + b cx + d
4 For each pair of functions, find the domain and range of gf(x).
a f(x) = 2x − 1 for x ∈ ℝ g(x) = 5x − 3 for x > 0
b f(x) = 7 − x for x < 5 g(x) = 3x − 2 for x > 2
c f(x) = 5x − 3 for x ∈ ℝ g(x) = x 2 − 3 for x > 5
d f(x) = 2x + 7 for x > − 4 g(x) = 9 − x 2 for 3 ø x ø 3
5 f(x) = x 2 + 4 for x < 0 g(x) = 3 − x 4 for 0 ø x < 8
a Find gf(x). Simplify your answer fully. [2]
b State the domain and range of gf(x) [3]
c Sketch gf(x). [2]
[Total: 7]
REFLECTION
How can you check that you have a good understanding of functions and compositions of functions?
When you have secure knowledge of these topics, it means that you will be able to use them in unfamiliar areas and questions.
On a scale of 1–5, how confident do you feel about functions and compositions?
On a scale of 1–5, how confident do you feel about finding the domains and ranges of compositions?
What can you do if you are not confident about these topics?
1.4 Modulus functions
RECALL AND CONNECT 2
Solve the following pair of simultaneous equations.
y = x − 7
y = x 2 − 9
1 Solve each equation for x
|4x + 7| = 15
2 Solve each pair of simultaneous equations.
3 f(x) = |1 − 2x| for x ∈ ℝ g(x) = 5 + x for x ∈ ℝ
a State i fg(2) [2] ii gf(2). [2]
b Solve gf(x) = 10. [3] [Total: 7]
1.5 Graphs of y = |f(
x)|
RECALL AND CONNECT 3
where f(x) is linear
Find the range of each of the following functions.
a f : x → 2 − 3x for 0 ø x ø 5
b g : x → |2 − 3x| for 0 ø x ø 5
c h : x → 2 − 3 | x| for 0 ø x ø 5
1 a Sketch y = |2x −
Sketch
2 Sketch each pair of functions and state the number of solutions to the equation f(x) = g(x).
On each sketch, label the lines and the points of intersection with the axes.
a f(x) = 3x + 2 g(x) = |2x − 6|
b f(x) = |4x + 2| g(x) = |x + 5|
c f(x) = |x + 3| + 1 g(x) = |2x + 1| −
UNDERSTAND THIS TERM
• Modulus
3 a i Sketch f(x) = |5x + 2| for 1 < x < 3
ii State the range of f(x).
b i Sketch f(x) = |3 − 2x | − 2 for 1 < x < 5
ii State the range of f(x)
c i Sketch f(x) = 3 − |3x + 8| for 2 < x < 6.
ii State the range of f(x)
4 a Sketch the graph of y = |3x − 5| showing the points where the graph meets the axes. [2]
b State the number of solutions to |3x − 5 | = 5 − x [1]
[Total: 3]
REFLECTION
In part b of Question 4, the command word is ‘state’. What should you remember when you see the word ‘state’? What hint might this command word give you about how to approach the question?
When you read a question, it can be a good idea to highlight all the command words, key words and specific instructions.
5 The diagram shows the graph of y = |f(x)|, where f(x) is a linear function. Find the two possible expressions for f(x). [3]
[Total: 3]
6 a Draw and label a sketch of the graph of y = |2x − 8| + 3. [3]
b It is given that the equation |2x − 8| + 3 = a − x has two solutions.
By considering the graphs of y = |2x − 8| + 3 and y = a − x, state an inequality for a [2]
[Total: 5]
1.6 Inverse functions
1 Which of the following functions have an inverse over the given domain?
A f(x) = 2x − 3 for x ∈ ℝ B f(x) = sin x for 0° ø x < 180 °
C f(x) = x 2 + 4x + 1 for x ∈ ℝ D f(x) = x 2 + 4x + 1 for x ù − 2
2 For each of the given functions, f(x), find the inverse function, f−1(x).
a f(x) = 8 x − 3 for x ∈ ℝ, x > 3
b f(x) = √ 2x + 5 + 3 for x ∈ ℝ, x ù 5 2
c f(x) = 3x + 2 2x − 5 for x ∈ ℝ, x ≠ 5 2
3 f(x) = (x + 2) 2 − 11 for 2 ø x < 3
Find f −1(x), stating its domain and range. [5]
[Total: 5]
4 f(x) = √ x − 5 + 3 for x > 5 g(x) = 4x + 8 for x > 0
a Find gf(x), stating its domain and range. [5]
b Hence find (gf) −1(x), stating its domain and range. [3]
[Total: 8]
1.7 The graph of a function and its inverse
1 f(x) = 3x − 7 for 1 ø x ø 5
On the same axes, sketch the graphs of y = f(x) and y = f −1(x), stating any points where the lines intersect the coordinate axes.
2 a On a copy of the axes, plot f(x) = (x − 1) 2 − 11 for x ù 1.
b Draw the graphs of
i y = x
ii y = f−1(x).
c Hence state the solution to f(x) = f −1(x)
3 a Sketch f(x) = x 2 − 4x, ∈ ℝ, x ù 0.
b Sketch f −1(x)
4 A function, g, is defined by g : x → 1 4 (x − p) 2 + q for x ø p
a Find g −1(x), stating its domain. [3]
b It is given that g(x) = g −1(x) exists and g(x) ù g −1(x)
i State the relationship between p and q. [1]
ii For the case p = 2, sketch on the same axes g(x) and g−1(x), clearly stating the value of the intercepts with axes. [4]
[Total: 8]
5 It is given that f(x) = 2(x − 3) 2 + 5 for x ø k
a State the greatest value of k for which f has an inverse. [1]
b For this k
i state the range of f(x) [1]
ii find f −1(x) and the domain and range [3]
iii sketch f(x) and f −1(x) on the same axes 0 ø x ø 10, 0 ø y ø 10 and state why f(x) = f −1(x) has no solutions. [4]
[Total: 9]
SELF-ASSESSMENT CHECKLIST
Let's revist the Knowledge and Exam skills focus for this chapter. Decide how confident you are with each statement.
Now I can
1understand functions and their domain and ranges
2form and use composite functions
Show it Needs more work Almost there Confident to move on
Make up a function and fix the domain. Try to find the range of your function. Use graphing software to check if you found the correct range for your given domain.
Write down two functions f(x) and g(x) and find the composite functions fg(x) and gf(x). Give your functions to a partner and challenge them to find the composite functions. Compare and check your answers.
SAMPLE
3sketch the graph of a modulus function and solve equations with modulus functions
4find the inverse of one–one functions
Write down the steps for sketching the graph of a modulus function.
Explain to a partner how to solve a modulus function equation graphically or algebraically.
Imagine a classmate has missed the lesson on inverse functions. Write some notes for them to explain how to find the inverse of a function. Include an example in your notes.
Compare your notes with a partner and check that you have included all the necessary steps.
CONTINUED
5sketch graphs to show the relationship between a function and its inverse
6show that I understand the command word ‘state’ and can answer ‘state’ questions
7show that I understand the command word ‘sketch’ and can put the correct key values on my sketch for this topic.
Write down a function of the form f(x) = (x + a) 2 + b and choose a domain that makes the function one–one. Sketch a graph of the function and its inverse. Find an expression for the inverse function and use graphing software to check your sketches.
Write a question using the command word ‘state’. Swap with a partner and answer each other’s questions. Swap back and check the answer.
Create a function of the form y = |ax + b| + c
Ask a partner to sketch the function and then check their answer:
• Is the sketch of y = |ax + b| + c drawn correctly?
• Is the y-intercept labelled?
• Is the point where it meets the x-axis labelled?
More digital questions on this topic can be found on Cambridge GO. Look inside the front cover for more information.