1: Functions1: Functions

““Teach A Level Maths”Teach A Level Maths”

Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules

Functions

e.g. and are functions.12)( xxf xxg sin)(

A function is a rule , which calculates values of for a set of values of x.

)(xf

is often replaced by y.)(xf

Another Notation

12: xxf 12)( xxfmeans

is called the image of x)(xf

Functions

)(f 12 xxx )(xf-

A few of the possible values of x

-

-

We can illustrate a function with a diagram

The rule is sometimes called a mapping.

Functions

We say “ real ” values because there is a branch of mathematics which deals with

numbers that are not real.

A bit more jargonTo define a function fully, we need to know the values of x that can be used.

The set of values of x for which the function is defined is called the domain.

In the function any value can be substituted for x, so the domain consists of

all real values of x

2)( xxf

means “ belongs to the set ”

So, means x belongs to the set of real numbers

x

stands for the set of all real numbers

We write x

Functions

What is a real number?

The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as and the 2;

or, a real number can be given by an infinite decimal representation, such as 2.48717..., where the digits continue in some way;

or, the real numbers may be thought of as points on an infinitely long number line.Real numbers measure continuous quantities such as weight, height, speed, time.

Functions

Tip: To help remember which is the domain and which the range, notice that d comes

before r in the alphabet and x comes before y.

0)( xf

If , the range consists of the set of

y-values, so

)(xfy

e.g. Any value of x substituted into gives a positive ( or zero ) value.

2)( xxf

So the range of is2)( xxf

The range of a function is the set of values given by .

)(xf)(xf

domain: x-values

range: y-values

Functions

The range of a function is the set of values given by the rule.

domain: x-values

range: y-values

The set of values of x for which the function is defined is called the domain.

Functions

Solution: The quickest way to sketch this quadratic function is to find its vertex by differentiating.

142 xxy 2 4 0dy

xdx

2 and 5x y

14)( 2 xxxfe.g. 1 Sketch the function where

and write down its domain and range.

)(xfy

)5,2( so the vertex is .

Functions

so the range is

5y

So, the graph of is 142 xxy

The x-values on the part of the graph we’ve sketched go from - to + . . . BUT we could have drawn the sketch for any values of x.

( y is any real number greater than, or equal to, - )

BUT there are no y-values less than -, . . .

)5,2( x

142 xxy

domain:

So, we get ( x is any real number )

x

Functions

3 xy

domain: x-values

range: y-values3x 0y

e.g.2 Sketch the function where .Hence find the domain and range of .

3)( xxf)(xfy )(xf

0

3

so the graph is:

( We could write instead of y )

)(xf

Solution: is a translation from ofxy xfy =

Functions

32x

1y

Asymptotes: if a function approaches a particular x or y value then the equation of this value is called an asymptote.Notice how each value of x maps (links) to ONLY ONE value of y .This is the definition of a function. If each value of x maps to MORE than one value of y it is NOT a function

Functions

32x

1y

As you cannot divide by zero x – 2 0Domain : x 2 Vertical asymptote at x = 2

The function approaches the line x = 2 but never meets it

Horizontal asymptote at y = 3. The graph approaches the line y = 3 but never meets it.

Range : y 3 or y<3 and y>3

Functions

SUMMARY

• To define a function we need a rule and a set of values.

)(xfy • For ,

the x-values form the domain

2)( xxf 2: xxf

• Notation:

means

the or y-values form the range)(xfe.g. For ,

the domain isthe range is or

2)( xxf

0y0)( xfx

Functions

(b) xy sin3xy(a)

Exercise

For each function write down the domain and range

1. Sketch the functions where

xxfbxxfa sin)()()()( 3 and

Solution:

)(xfy

range: 11 y

domain: x

domain: x

range: y

Functions

3xSo, the domain is

03x 3x

We can sometimes spot the domain and range of a function without a sketch.

e.g. For we notice that we can’t square root a negative number ( at least not if we want a real number answer ) so,

3)( xxf

x must be greater than or equal to zero.

3xThe smallest value of is zero.Other values are greater than zero.So, the range is

0y

Functions

xy sin13 xy

One-to-one and many-to-one functions

Each value of x maps to only one value of y . . .

Consider the following graphs

Each value of x maps to only one value of y . . .

BUT many other x values map to that y.

and each y is mapped from only one x.

and

Functions

One-to-one and many-to-one functions

is an example of a one-to-one function

13 xy is an example of a many-to-one function

xy sin

xy sin13 xy

Consider the following graphs

and

Functions

Other one-to-one functions are:

xy

1 xy 4

and

Functions

Here the many-to-one function is two-to-one ( except at one point ! )

432 xxy 863 23 xxxy

Other many-to-one functions are:

This is a many-to-one function even though it is one-to-one in some parts.

It’s always called many-to-one.

Functions

1 xy

This is not a function. Functions cannot be one-to-many.

We’ve had one-to-one and many-to-one functions, so what about one-to-many?One-to-many relationships do exist BUT, by definition, these are not functions.

is one-to-many since it gives 2 values of y for all x values greater than 1.

1,1 xxye.g.

So, for a function, we are sure of the y-value for each value of x. Here we are not sure.