1 of 43

P2 Chapter 2 :: Functions & Graphs

www.dilanmaths.com�

2 of 43

Chapter Overview

1:: The Modulus Function

2:: Mappings vs Functions, Domain and Range

3:: Composite Functions

4:: Inverse Functions

3 of 43

1 :: The Modulus Function

1

2

3

4

5

6

7

8

4 of 43

Examples

?

5 of 43

Modulus Graphs

	-2	-1	0	1	2
	2	1	0	1	2

?

6 of 43

Modulus Graphs

Fro Tip: I like to sketch the non-modulus graph first with a dotted line.

As you would have done in Pure Year 1, sketch a line for each side of the equation, so that we can use the points of intersection.

7 of 43

Test Your Understanding

?

8 of 43

Exercise 2A

Pearson Pure Mathematics 3

Page 14

9 of 43

What is a mapping?

-1

0

1.7

2

...

3.1

-1

1

4.4

5

...

7.2

A mapping is something which maps one set of numbers to another.

Inputs

Outputs

🖉 The domain is the set of possible inputs.

🖉 The range is the set of possible outputs.

Also notice that one input might map to multiple outputs, or multiple inputs to one output.

Notice also that not all values in the set of inputs necessarily have a mapping to a value in the set of outputs.

10 of 43

What is a function?

A function is: a mapping such that every element of the domain is mapped to

exactly one element of the range.

Notation:

Function?

?

Yes

No

Yes

No

Yes

No

Yes

Fro Tip: Use the ‘vertical ray test’. If a vertically fired ray can hit the curve multiple times, it is NOT a function.

No

Yes

11 of 43

One-to-one vs Many-to-one

While functions permit an input only to be mapped to one output, there’s nothing stopping multiple different inputs mapping to the same output.

Many-to-one

function

Multiple inputs can map to the same output.

2

-2

4

Type

Description

Example

One-to-one

function

Each output has one input and vice versa.

2

3

5

7

4

9

?

You can use the ‘horizontal ray test’ to see if a function is one-to-one or many-to-one.

12 of 43

Further Examples

It is often helpful to sketch the function to reason about the range.

a

b

c

?

13 of 43

Piecewise Functions

A ‘piecewise function’ is one which is defined in parts: we can use different rules for different intervals within the domain.

The filled/unfilled circles have the same meaning as with inequalities on a number line – unfilled indicates not included.

a

b

?

14 of 43

Test Your Understanding

Edexcel C4 June 2012 Q6a

Edexcel C4 June 2010 Q4d

Hint: Identify the minimum point first, as this may or may not affect the range.

Extra Hint: Carefully consider the stated domain.

Notice the range doesn’t include 2, as the line never reaches the asymptote.

?

15 of 43

Exercise 2B

Pearson Pure Mathematics 3

Pages 18-19

16 of 43

Summary of Domain/Range

It is important that you can identify the range for common graphs, using a suitable sketch:

Be careful in noting the domain – it may be ‘restricted’, which similarly restricts the range. Again, use a sketch!

?

17 of 43

Just for your interest…

Consequence 1

Consequence 2

A bit of Computer Science!

In many programming languages, we can pass functions as the parameters of a method, when a variable is allowed to have a function as its value.

function map(f, a) {� let b be a new list

for(i from 1 to size(a)) {

b_i = f(a_i)

}

return b

}

18 of 43

Composite Functions

Sometimes we may apply multiple functions in succession to an input.

These combined functions are known as a composite function.

19 of 43

Examples

?

20 of 43

Further Examples

?

a

b

21 of 43

Test Your Understanding

Edexcel C4 June 2013(R) Q4

Edexcel C4 June 2012 Q6

? b

? d

?

22 of 43

Exercise 2C

Pearson Pure Mathematics 3

Pages 22-23

Extension

?

1

2

23 of 43

Inverse Functions

Explain why the function must be one-to-one for an inverse function to exist:

If the mapping was many-to-one, then the inverse mapping would be one-to-many. But this is not a function!

?

This has appeared in exams before.

24 of 43

25 of 43

Graphing an Inverse Function

?

26 of 43

Example

a

b

c

d

?

27 of 43

Further Example

From graph, we can see we only want positive solution.

a

c

b

?

Note: There was once an exam question based on this principle.

?

28 of 43

Test Your Understanding

Edexcel C4 June 2012 Q6

?

29 of 43

Exercise 2D

Pearson Pure Mathematics 3

Pages 26-27

30 of 43

Sketch >

31 of 43

Test Your Understanding

Edexcel C4 June 2012 Q4

Sketch >

32 of 43

Further Test Your Understanding

b

?

a

33 of 43

Exercise 2E

Pearson Pure Mathematics Year 2/AS

Pages 30-32

Extension

1

?

34 of 43

Combining Transformations

RECAP:

Affects which axis?

What we expect or opposite?

Opposite

What we expect

?

There is nothing new here relative to Year 1, except that you might have to do more than one transformation…

Start with the points you have coordinates of first!

?

35 of 43

Combining Transformations

? b

? c

36 of 43

Test Your Understanding

C4 June 2011 Q3

? b

? a

37 of 43

?

38 of 43

Exercise 2F

Pearson Pure Mathematics 3

Pages 34-35

39 of 43

Solving Modulus Problems

It is often helpful to sketch the graph in stages as we apply more transformations:

-1 is ‘inside’ function so translate 1 right.

-2 is outside modulus function so translate 2 down.

a

40 of 43

Solving Modulus Problems

b

?

41 of 43

Solving Modulus Problems

Fro Note: Only the modulus bit is negated, not the whole equation.

?

c

42 of 43

Test Your Understanding

C4 June 2008 Q3

a

b

?

43 of 43

Exercise 2G

Pearson Pure Mathematics 3

Pages 38-39

Extension

?