C3 Chapter 2 Functions

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

Last modified: 10th September 2014

What is a mapping?

f(x) = 2x + 1

-1

0

1.72

...

3.1

-1

1

4.45

...

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.

What is a mapping?We can also illustrate a mapping graphically, where the axis is the input and the axis is the output .

𝑥

𝑦where

𝑦=𝑓

(𝑥)

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:

𝑓 (𝑥 )=2𝑥+1 𝑓 :𝑥→2 𝑥+1

x

y

No Yes

x

y

No Yes

f(x) = 2x

No Yes

Domain: all real numbers f(x) = √x Domain:

No Yesf(4) = 2 but f(4) = -2 also.This violates the definition of a function.

For each input ( value), we only get one output ( value)

For each value of (except 0), we get two values of !

Function?

?

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-onefunction

Multiple inputs can map to the same output.

2

-2

4

f(x) = x2

e.g. f(2) = 4f(-2) = 4

Type Description Example

One-to-onefunction

Each output has one input and vice versa.

23

5

7

4 9

f(x) = 2x + 1

? ?

? ?

Example

Domain: 𝑥∈ℝ…the set of real numbers

Range:We can use any real number as the input!

𝑓 (𝑥 )≥0The output has to be positive, since it’s been squared.

?

?

Type: Many-to-one ?

𝑓 (𝑥 )=𝑥2 Sketch:𝑥

𝑦

?is an element of

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.

?

?

Type: One-to-one ?

Sketch:

𝑥

𝑦

?

ExerciseDetermine 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

Functions.t.

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

? ? ?

? ? ?

? ? ?

Check Your Understanding So Far

A function is: a mapping such that every element of the domain is mapped to exactly one element of the range.

The domain is: the set of possible inputs.

The range is: the set of possible outputs.

Give an example of a one-to-one function:Any increasing or decreasing function, such as any linear function (e.g. ), exponential, . Also …

Give an example of a cubic equation which is one-to-one: will do!

Give an example of a cubic equation which is many-to-one: (think about its graph)

What is the range of ?

?

?

?

?

?

?

?

Composite Functions

𝑥 𝑓 (𝑥 ) 𝑔𝑓 (𝑥 )

𝑓 𝑔

𝑔𝑓

We can apply another function, say , to the output of .

Bro Tip: Think of as . This will allow you to remember that we apply first and then .

Examples

Let , and .What is…

? ?

?

?

?

?

?

?

Quickfire ExamplesDo in your head!

? ? ? ?? ? ? ?? ? ? ?? ? ? ?? ? ? ?? ? ? ?

1

2

3

4

5

6

The opposite: determining sequence of functions

Sometimes you’re given a composite function, and have to determine what functions it is composed of.

e.g. If , , Find in terms of the following functions:

Bro Tip: We know and are going to come into it somehow. And given , it looks like we do first before .

?

?

?

Exercise 2DGiven that and , find expressions for the functions:

If and , find the number(s) such that .

The functions and are defined by , and . Find in terms of the functions:a) b) c) d) e) f) g)

If and , prove that

1

3

5

7

?????

?

??

?

??

??

?

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!

𝑓 (𝑥 )

𝑥 𝑦

𝑓 −1 (𝑥 )

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

?

Finding the Inverse FunctionIf is defined as a) Find the range of .b) Calculate c) Sketch the graphs of both functions.d) State the domain and range of .e) What do you notice about the graphs of the two functions, and

the domain/range of the two functions?

Start with and make the subject, before swapping and .

𝑥

𝑦

𝑦=𝑔 (𝑥)

𝑦=𝑔−1(𝑥)

2

2

Domain:

Range:

The domain and range have swapped.Since and swaps to find the inverse, it’s equivalent to a reflection in the line .

a

b

c d

e

?

??

?

?

Quickfire Inverses

Original Inverse

?????

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

(Can you think of others?)

Test Your Understanding

A function is defined as

a) State a suitable domain.b) Find c) Find d) State the range of

Domain:

So

The domain of is the range of .So

a

b

c

Q

Find Sketch and on the same axis.Q

So

𝑥

𝑦 𝑥=1

−3

𝑦=1−3

??

Exercise 2E

Q1a, c, d, e, Q2, Q3, Q4a, c, e, Q6, Q7