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www.le.ac.uk Moduli functions Department Name University of Leicester

Www.le.ac.uk Moduli functions Department Name University of Leicester

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Page 1: Www.le.ac.uk Moduli functions Department Name University of Leicester

www.le.ac.uk

Moduli functions

Department NameUniversity of Leicester

Page 2: Www.le.ac.uk Moduli functions Department Name University of Leicester

Content

Inverse functions

Modulus function

Inequalities with moduli

Introduction

Page 3: Www.le.ac.uk Moduli functions Department Name University of Leicester

Introduction

Modulus functions

Inequalities with moduli

Next

Introduction

Page 4: Www.le.ac.uk Moduli functions Department Name University of Leicester

Modulus

The modulus (or absolute) function is written as and produces the graph

Next

Modulus functions

Inequalities with moduli

Introduction

Page 5: Www.le.ac.uk Moduli functions Department Name University of Leicester

Modulus of other functions

𝑦=sin 𝑥

𝑦=|sin𝑥|

The modulus effectively reflects all the values below the horizontal axis in the line

Next

Modulus functions

Inequalities with moduli

Introduction

Page 6: Www.le.ac.uk Moduli functions Department Name University of Leicester

Plotting

Enter numbers between -10 and 10:y = | x + |

2 4 6 80

2

-2

-1

0

1

3

-3

-4

1 3 5 7-4 -2-5 -3 -1-6

Next

Modulus functions

Inequalities with moduli

Introduction

Plot without modulus sign

Plot with modulus sign

Clear

Plot without modulus sign

Plot with modulus sign

Page 7: Www.le.ac.uk Moduli functions Department Name University of Leicester

Question

2 4 6 80

2

4

6

0

1

3

5

1 3 5 7-4 -2-5 -3 -1-6

Select Answe

r

Select Answe

r

Select Answe

r

Which of these is the graph of ?

Select Answe

r

Modulus functions

Inequalities with moduli

Introduction

Page 8: Www.le.ac.uk Moduli functions Department Name University of Leicester

Question

20

2

4

6

0

1

3

5

1 3-4 -2-5 -3 -1-6

Which of these is the graph of ?

Select Answe

r

Select Answe

rSelect Answe

r

Select Answe

r

Modulus functions

Inequalities with moduli

Introduction

Page 9: Www.le.ac.uk Moduli functions Department Name University of Leicester

Question

Modulus functions

Inequalities with moduli

Introduction

Which of these is the graph of ?

Page 10: Www.le.ac.uk Moduli functions Department Name University of Leicester

Inequalities with moduli

2 4 6 80

2

-2

-1

0

1

3

-3

-4

1 3 5 7-4 -2-5 -3 -1-6

| x + | < | x + |(This line is in blue) (This line is in pink)

Next

Modulus functions

Inequalities with moduli

Introduction

Plot the 2 lines

Clear

Shade valid region

Page 11: Www.le.ac.uk Moduli functions Department Name University of Leicester

Question

2 4 6 80

2

4

0

1

3

5

1 3 5 7-4 -2-5 -3 -1-6

𝑦=¿ 𝑥+4∨+1

𝑥<−0.2

𝑥<0−4<𝑥<0

or

𝑥>3

𝑥>−0.2

−0.2<𝑥<3−0.2<𝑥<1

or

The graphs of and are shown. What is the solution to ?

6

𝑦=¿ 4 𝑥 – 4∨¿

Modulus functions

Inequalities with moduli

Introduction

Page 12: Www.le.ac.uk Moduli functions Department Name University of Leicester

Conclusion

Modulus functions

Inequalities with moduli

Next

Introduction

You should now be able to:

Recognise and draw the graphs of functions with moduli in them.

Solve inequalities with moduli in them.

Page 13: Www.le.ac.uk Moduli functions Department Name University of Leicester