IGCSE FM Domain/Range

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 14th October 2015

Objectives: The specification:

OVERVIEWThis is a IGCSE FM topic only (not C1 – you don’t see domain/range until C3!)

#1: Understanding of functions #2: Domain/Range of common functions (particularly quadratic and trigonometric)

#3: Domain/Range of other functions

#4: Constructing a function based on a given domain/range.

RECAP :: Functions

𝑓 (𝑥)=2 𝑥

f𝑥 2 𝑥Input Output

A function is something which provides a rule on how to map inputs to outputs.From primary school you might have seen this as a ‘number machine’.

Input OutputName of the function (usually or )

Check Your Understanding

𝑓 (𝑥)=𝑥2+2What does this function do?It squares the input then adds 2 to it.

What is ?f(3) = 32 + 2 = 11

What is ?f(-5) = 27

If , what is ? So

Q1

Q2

Q3

Q4

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Algebraic Inputs

If you change the input of the function (), just replace each occurrence of in the output.

If what is:

𝑓 (𝑥−1 )= (𝒙−𝟏 )+𝟏=𝒙If what is:

𝑓 (𝑥−1 )= (𝒙−𝟏 )𝟐−𝟏?

???

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?If what is:

𝑓 (𝑥−1 )=𝟐 (𝒙 −𝟏 )?

??

Test Your Understanding

If , solve

2 𝑥2+1=51

If , determine:(a) (b) (c)

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A

B

Exercise 1[AQA Worksheet] . Work out when

[AQA Worksheet] .If , determine the value of .

[AQA Worksheet] Show that

[June 2012 Paper 2] for all values of . Solve

[AQA Set 2] The function is defined as

(a) Work out the value of

(b) Work out the value of (b) Solve

(only 2 within domain)

(which is in domain)

If determine:(a) (b) (c) (d)

(e) Solve

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(exercises on provided sheet)

Domain and Range

𝑓 (𝑥 )=𝑥2

-1

0

1.72

...

3.1

1

0

2.894

...9.61

Inputs

Outputs

! The domain of a function is the set of possible inputs.

! The range of a function is the set of possible outputs.

Example

Domain: for all

Range:

We can use any real number as the input!

𝑓 (𝑥 )≥0Look at the values on the graph.The output has to be positive, since it’s been squared.

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𝑓 (𝑥 )=𝑥2 Sketch:𝑥

𝑦

B Bro Tip: Note that the domain is in terms of and the range in terms of .

Test Your Understanding

𝑓 (𝑥 )=√𝑥

Domain: 𝑥≥0

Range:

Presuming the output has to be a real number, we can’t input negative numbers into our function.

𝑓 (𝑥 )≥0The output, again, can only be positive.

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Sketch:

𝑥

𝑦

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Function

Domain For all except 2

Range For all except 1

Function

Domain For all

Range

Mini-ExerciseIn pairs, work out the domain and range of each function.A sketch may help with each one.

Function

Domain For all

Range For all

Function

Domain For all

Range

Function

Domain For all except 0

Range For all except 0

1 23

Function

Domain For all

Range

4

Functions.t.

Domain

Range

Function

Domain

Range

8 9

? ? ?

? ?

? ?

5 Function

Domain For all

Range For all ?

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Range of Quadratics

A common exam question is to determine the range of a quadratic.

𝑥

𝑦 The sketch shows the function where .Determine the range of .

We need the minimum point, since from the graph we can see that (i.e. ) can be anything greater than this.

The minimum point is thus the range is:

(note the rather than )

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3

An alternative way of thinking about it, once you’ve completed the square, is that anything squared is at least 0. So if is at least 0, then clearly is at least 3.

Test Your Understanding

𝑥

𝑦The sketch shows the function where .Determine the range of .

Therefore

𝑥

𝑦The sketch shows the function where .Determine the range of .

Therefore

(1 ,−9 )

(2,25 )

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Range for Restricted Domains

Some questions are a bit jammy by restricting the domain. Look out for this, because it affects the domain!

Determine the range of .

𝑥

𝑦

−3 −1 1

Notice how the domain is .

When Sketching the graph, we see that when , the function is increasing.Therefore when

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Test Your Understanding

Determine the range of .

When As decreases from -2, is increasing. Therefore:

𝑥

𝑦

−2 ?

Determine the range of .

When When

Range:

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Range of Trigonometric Functions

90 ° 180 ° 270 ° 360 °

Domain Range

For all (i.e. unrestricted)

Suppose we restricted the domain in different ways.Determine the range in each case (or vice versa). Ignore angles below 0 or above 360.

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Range of Piecewise Functions

It’s a simple case of just sketching the full function.

The sketch shows the graph of with the domain

Determine the range of .

Range:Graph ? Range ?

Test Your Understanding

The function is defined for all :

Determine the range of .

Range:

Graph ? Range ?

Exercise 2Work out the range for each of these functions.(a) for all

(b)

(c)

(a) Give a reason why is not a suitable domain for .It would include 3, for which is undefined.(b) Give a possible domain for

The range of is Work out and .

[Set 1 Paper 2] (a) The function is defined as:

The range of is Work out the value of .

(b) The function is defined as for all .(i) Express in the form

(ii) Hence write down the range of .

[June 2012 Paper 1] for all values of .(a) What is the value of ?

(b) What is the range of ?

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(exercises on provided sheet)

Exercise 2[Jan 2013 Paper 2]

(a) What is the range of ?

(b) You are given that .Work out the value of .

By completing the square or otherwise, determine the range of the following functions:(a) for all

Range: (b) for all

Range:

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Here is a sketch of for all , where is a constant. The range of is . Work out the value of .

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(exercises on provided sheet)

Exercise 2

The straight line shows a sketch of for the full domain of the function.(a) State the domain of the function.

(b) Work out the equation of the line.

is a quadratic function with domain all real values of . Part of the graph of is shown.(a) Write down the range of .

(b) Use the graph to find solutions of the equation .

(c) Use the graph to solve .

9 10

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(exercises on provided sheet)

Exercise 2The function is defined as:

Work out the range of .

The function has the domain and is defined as:

Work out the range of .[June 2012 Paper 2] A sketch of for domain is shown.The graph is symmetrical about . The range of is .Work out the function .

11 13

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(exercises on provided sheet)

Constructing a function from a domain/rangeJune 2013 Paper 2

𝑥

𝑦

1 5

3

11

What would be the simplest function to use that has this domain/range?A straight line! Note, that could either be going up or down (provided it starts and ends at a corner)

What is the equation of this?

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Constructing a function from a domain/rangeSometimes there’s the additional constraint that the function is ‘increasing’ or ‘decreasing’. We’ll cover this in more depth when we do calculus, but the meaning of these words should be obvious.

is a decreasing function with domain and range .

𝑦

4 6

7

19

𝑥

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Domain is . Range . is an increasing function.

Domain is . Range . is a decreasing function.

Domain is . Range . is an increasing function.

Domain is . Range . is a decreasing function.

Domain is . Range . is a decreasing function.

Exercise 3

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(exercises on provided sheet)