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P3 Chapter 2 Functions

www.dilanmaths.com

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What is a mapping?

f(x) = 2x + 1

-1

0

1.7

2

...

3.1

-1

1

4.4

5

...

7.2

A mapping is something which maps an input value to an output value.

Inputs

Outputs

🖉 The domain is the set of possible inputs.

🖉 The range is the set of possible outputs.

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What is a mapping?

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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:

x

y

🗶

✔

No

Yes

x

y

🗶

✔

No

Yes

f(x) = 2^x

🗶

✔

No

Yes

Domain: all real numbers

f(x) = ±√x

🗶

✔

No

Yes

f(4) = 2 but f(4) = -2 also.

This violates the definition of a function.

Function?

?

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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

f(x) = x²

e.g. f(2) = 4

f(-2) = 4

Type

Description

Example

One-to-one

function

Each output has one input and vice versa.

2

3

5

7

4

9

f(x) = 2x + 1

?

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Example

Domain:

…the set of real numbers

Range:

We can use any real number as the input!

The output has to be positive, since it’s been squared.

?

Type:

Many-to-one

?

Sketch:

?

is an element of

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Test Your Understanding

Domain:

Range:

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

The output, again, can only be positive.

?

Type:

One-to-one

?

Sketch:

?

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Exercise

Determine the domain, range and type of function/mapping, as well as a quick sketch of the graph.

Function
Domain
Range
Type	One-to-one

Function
Domain
Range
Type	One-to-one

Function
Domain
Range
Type	One-to-one

Function
Domain
Range
Type	One-to-one

1

2

3

4

Function
Domain
Range
Type	Many-to-one

5

Function
Domain
Range
Type	Not a function/many-to-many

Function
Domain
Range
Type	Many-to-one

Function
Domain
Range
Type	Many-to-one

7

6

Function
Domain
Range
Type	Many-to-one

8

9

?

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Check Your Understanding So Far

?

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Composite Functions

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Examples

?

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Quickfire Examples

Do in your head!

?

1

2

3

4

5

6

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The opposite: determining sequence of functions

?

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Exercise

1

3

5

7

?

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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!

The inverse of a function maps the output values back to the input values.

?

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Finding the Inverse Function

a

b

c

d

e

?

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Quickfire Inverses

Original	Inverse

?

A self-inverse function is a function that is the same as its inverse.

(Can you think of others?)

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Test Your Understanding

a

b

c

Q

?