The height of a ball thrown vertically upwards with initial speed 20ms -1 after time t is h = 20t...

The height of a ball thrown vertically upwards with initial speed 20ms-1 after time t is h = 20t – 5t2.

The formula gives h in terms of t, and assigns to each value of t between 0 and 4 (why?) a unique value of h between… (what?)

The idea of a functionFUNCTIONS

The height of a ball thrown vertically upwards with initial speed 20ms-1 after time t is h = 20t – 5t2.

The formula gives h in terms of t, and assigns to each value of t between 0 and 4 (why?) a unique value of h between… (what?)

Solution

Why?When h = 0, the ball must be at the point from which it was thrown.So, 0 = 20t – 5t2

0 = 5t(4 – t) → t = 0 or t = 4.

What?Maximum height occurs when = 0.

= 20 – 10t

20 – 10t = 0 when t = 2

So, when t = 2, h = 20(2) – 5(2)2 = 20m. So, h ranges from 0m to 20m.

The idea of a functionFUNCTIONS

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This is an example of a function: notation h(t).

Functions are examples of a wider class called mappings.

The idea of a functionFUNCTIONS

For example,

Why fly to Geneva in January?

Several people arriving at Geneva airport from London were asked the main purpose of their visit. Their answers were recorded.

David

Joanne Skiing

Jonathan Returning home

Louise To study abroad

Paul Business

Shamaila

Karen

The language of functionsFUNCTIONS

This is an example of a mapping.

A mapping is any rule which associates two sets of items. In this example each of the names on the left is an object or input, and each of the reasons on the right is an image, or output.

The language of functionsFUNCTIONS

For a mapping to make sense or to have any practical application, the inputs and outputs must each form a natural collection or set.

The set of possible inputs (in this case, all the people who flew to Geneva from London in January) is called the domain of the mapping.

The set of possible outputs (in this case, the set of all possible reasons for flying to Geneva) is called the co-domain of the mapping.

The language of functionsFUNCTIONS

The seven people questioned in this example gave a set of four reasons, or outputs. These form the range of the mapping for this particular set of inputs.

The range of any mapping forms part or all of its co-domain.

The language of functionsFUNCTIONS

Notice that Jonathan, Louise and Karen are all visiting Geneva on business: each person only gave one reason for the trip, but the same reason was given by several people.

This mapping is said to be many-to-one.

A mapping can also be one-to-one, one-to-many or many-to-many.

The language of functionsFUNCTIONS

The relationship between the people and their UK passport numbers will be one-to-one.

The relationship between the people and their items of luggage is likely to be one-to-many, and that between the people and the countries they have visited in the last 10 years will be many-to-many.

The language of functionsFUNCTIONS

In Mathematics, many (but not all) mappings can be expressed using algebra. Here are some examples of Mathematical mappings.

For each of these examples:

Decide whether the mapping is one-to-one, many-to-many, one-to-many or many-to-one.

MappingsFUNCTIONS

Sets of numbersReal numbers ℝ, ℝ+, ℝ-

Rational Numbers ℚ, ℚ+, ℚ-

Integers ℤ, ℤ+, ℤ-

Natural Numbers ℕ

FUNCTIONS Sets of numbers

(ℝ) Real numbers

Irrational numbers: √2, π, e.

(ℚ) Rational numbers⅓, -3, 1.4, 0.25

(ℤ) Integers-3, -2, -1, 0, 1, 2, 3

(ℕ) Natural numbers 0,1,2,3,4…

Domain: integers Co-domain: real numbers

Objects Images

-1 3

0 5

1 7

2 9

3 11

General rule:

x 2x + 5

MappingsExample 1

Domain: integers Co-domain: real numbers

Objects Images

-1 3

0 5

1 7

2 9

3 11

General rule:

x 2x + 5

MappingsExample 1

One-to-one

Domain: integers Co-domain: real numbers

Objects Images

1.9

2 2.1

2.33

2.52

3 2.99

π

General rule:

Rounded Unroundedwhole numbers numbers

MappingsExample 2

Domain: integers Co-domain: real numbers

Objects Images

1.9

2 2.1

2.33

2.52

3 2.99

π

General rule:

Rounded Unroundedwhole numbers numbers

MappingsExample 2

One-to-many

Domain: real numbers Co-domain: real numbers

Objects Images

0

45 0

90 0.707

135 1

180

General rule:

x° sin x°

MappingsExample 3

Domain: real numbers Co-domain: real numbers

Objects Images

0

45 0

90 0.707

135 1

180

General rule:

x° sin x°

MappingsExample 3

many-to-one

Domain: quadratic Co-domain: real numbersequations with real rootsObjects Images

x2 – 4x + 3 = 0 0

x2 – x = 0 1

x2 – 3x + 2 = 0 2

3General rule:

ax2 + bx + c = 0 x =

x =

MappingsExample 4

-b - √(b2 – 4ac)2a

-b - √(b2 – 4ac)2a

Domain: quadratic Co-domain: real numbersequations with real rootsObjects Images

x2 – 4x + 3 = 0 0

x2 – x = 0 1

x2 – 3x + 2 = 0 2

3General rule:

ax2 + bx + c = 0 x =

x =

For practice go to: www.supermathsworld.compassword: clvmathsexpert – functions 1 – mapping diagrams

MappingsExample 4

-b - √(b2 – 4ac)2a

-b - √(b2 – 4ac)2a

many-to-many

Mappings which are one-to-one or many-to-one are of particular importance, since in these cases there is only one possible image for every object.

Mappings of these types are called functions.

For example, x → x2 and x → cos x° are both functions, because in each case for any value of x there is only one possible answer.

The mapping of rounded whole numbers onto un-rounded numbers is not a function, since, for example, the rounded number 5 could be the object for any un-rounded number between 4.5 and 5.5.

There are several different equivalent ways of writing a function. For example, the function which maps x onto x2 can be written in any of the following ways.

• y = x2

• f(x) = x2

• f:x → x2

FunctionsFUNCTIONS

Read this as ‘f maps x onto x2‘

It is often represent a function graphically, as in the following examples, which also illustrate the importance of knowing the domain.

FunctionsFUNCTIONS

FunctionsExample 5

Sketch the graph of y = 3x + 2 when the domain of x is:

a) x ℝ

b) x ℝ+ (i.e. positive real numbers)

c) x ℕ

FunctionsAnswers

a) x ℝ b) x ℝ+ c) x ℕ

When the domain is ℝ, all values of y are possible. The range is therefore ℝ, also.

When x is restricted to positive values, all the values of y are greater than 2, so the range is y > 2.

In this case the range is the set of points {2, 5, 8, …}. These are clearly all of the form 3x + 2 where x is a natural number (0, 1, 2, …). This set can be written neatly as {3x + 2: x ℕ}.

The open circle shows that (0, 2) is not part of the line

y

x

y

x

y

x

FunctionsExample 6

Sketch the graph of the function y = f(x)

f(x) = x2 when the domain of x is 0 ≤ x < 3

FunctionsAnswer

Sketch f(x) = x2 0 ≤ x < 3y

x

The closed circle shows that (0, 0) is

part of the line

Restricting the domain has produced a one-to-one function here.

For practice go to: www.supermathsworld.compassword: clvmathsexpert – functions 2 – domain

FunctionsFUNCTIONS

When you draw the graph of a mapping, the x co-ordinate of each point is an input value, the y coordinate is the corresponding output value. The table below shows this for the mapping x → x2, or y = x2, and the figure shows the resulting points on a graph.

Input (x) Output (y) Point plotted

-2 4 (-2, 4)

-1 1 (-1, 1)

0 0 (0, 0)

1 1 (1, 1)

2 4 (2, 4)

y

x

If a mapping is a function, there is one and only one value of y for every value of x in the domain. Consequently, the graph of a function is a simple curve or line going from left to right, with no doubling back.

FunctionsFUNCTIONS

y = x2 y = ± √ 9 – x2

Is it a function?Question 1

Does this graph show a function?

Yes / No

x

y

Is it a function?Question 2

Does this graph show a function?

Yes / No

x

y

Is it a function?Question 3

Does this graph show a function?

Yes / No

x

y

Is it a function?Question 4

Does this graph show a function?

Yes / No

x

y

Is it a function?Question 5

Does this graph show a function?

Yes / No

x

y

Is it a function?Question 6

Does this graph show a function?

Yes / No

x

y

Is it a function?Question 7

Does this graph show a function?

Yes / No

x

y

Is it a function?Question 8

Does this graph show a function?

Yes / No

x

y

Is it a function?Question 9

Does this graph show a function?

Yes / No

x

y

Is it a function?Question 10

Does this graph show a function?

Yes / No

x

y

Is it a function?Answers

1. No (a one to many mapping)

2. Yes (a one to one mapping)

3. No (a one to many mapping)

4. Yes (a one to one mapping)

5. Yes (a one to one mapping)

6. No (a many to many mapping)

7. Yes (a one to one mapping)

8. Yes (a many to one mapping)

9. Yes (a one to one mapping)

10. Yes (a many to one mapping)

FunctionsMini - Review

What is the domainof a function?

FUNCTIONS

A rule that is a function…

A rule that is not a function…

Match each letter to the correct set of numbers.

Natural numbers

Real numbers

Rational numbers

Integers

ℝ

ℚ

ℤ

ℕ

Show two different ways to write the same function…