1: Functions1: Functions

© Christine Crisp

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

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

Functions

Module C3

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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 )(xf1

0

.

.

2.1...

.

1.5.

.

.

.

.

A few of the possible values of x

3.2...

2..

1..

3.

.

.

.

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 ”So, means x is any real number

x

stands for the set of all real numbers

We write x

Functions

0)( xf

If , the range consists of the set of y-values, so

)(xfy

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.

domain: x-values

range: y-values

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

2)( xxf

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

)(xf)(xf

So the range of is2)( xxf

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 completing the square.

142 xxy 2)2( xy 4 1

5)2( 2 xy

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

and write down its domain and range.

)(xfy

5

2This is a translation from of2xy

)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 5 to 1 . . . BUT we could have drawn the sketch for any values of x.

( y is any real number greater than, or equal to, 5 )

BUT there are no y-values less than 5, . . .

)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

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)(xf

e.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 domai

n: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 + 3 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

then,

)(f 3

Suppose and2)( xf x )(xg 3xFunctions of a Function

x is replaced by 3

Functions

)(xgand f

2)( 3

Suppose and2)( xf )(xg

)(f 1

then,

)(f 32)( 191

x 3xFunctions of a Function

x is replaced by 1x is replaced by )(xg

Functions

3x)(xgand f

is “a function of a function” or compound function.

f )(xg

2)( 3x

962 xx

f

2)( 3

Suppose and2)( xf )(xg

)(f 1

then,

)(f 32)( 191

x 3x

We read as “f of g of x” )(xgf

x is “operated” on by the inner function first.

is the inner function and the outer.)(xg )(xf

So, in we do g first. )(xgf

Functions of a Function

Functions

Notation for a Function of a Function

When we meet this notation it is a good idea to change it to the full notation.

is often written as . f )(xg )(xfg

does NOT mean multiply g by f.)(xfg

I’m going to write always !

f )(xg

Functions

Solution:

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

fxfg )((i) )(xg

Functions

x

1)(xg

xg

1)(

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfe.g. 1 Given that and find x x

Solution: fxfg )((i)

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

x

f

Functions

x

1)(xgSolution: fxfg )((i)

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

f 2

2

x

1

Functions

Solution: fxfg )((i)

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

f

x

1)(xg 22

x

1

gxgf )((ii) )(xf

212

x

Functions

)(xf

Solution: fxfg )((i)

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

f

x

1)(xg 22

x

1

gxgf )((ii) )(g 22 x

212

x

Functions

22 x)(xf

Solution:

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

f

x

1)(xg 22

x

1

gxgf )((ii) )(g

)(xgfN.B. is not the same as )(xfg

fxfg )((i)

2

12

x

212

x

Functions

212

x

2

12

x

Solution:

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

f

x

1)(xg 22

x

1

gxgf )((ii) )(xf )(g 22 x

)(xgfN.B. is not the same as )(xfg

fxfg )((i)

Functions

2

12

x

Solution: fxfg )((i)

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

f

x

1)(xg 22

x

1

)()( xffxff(iii)

212

x

gxgf )((ii) )(xf )(g 22 x

Functions

22 x

gxgf )((ii) )(xf )(g 22 x

Solution: fxfg )((i)

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

f

x

1)(xg 22

x

1

)()()( fxffxff(iii)

212

x

2

12

x

Functions

64 24 xx

gxgf )((ii) )(xf )(g 22 x

Solution: fxfg )((i)

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

f

x

1)(xg 22

x

1

)()()( fxffxff(iii) 22 x 2)( 2 22 x

212

x

2

12

x

Functions

gxgf )((ii) )(xf )(g 22 x

Solution: fxfg )((i)

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

f

x

1)(xg 22

x

1

)()()( fxffxff(iii) 22 x 2)( 2 22 x

212

x

)()( xggxgg(iv)64 24 xx

2

12

x

Functions

1

x1x

1

gxgf )((ii) )(xf )(g 22 x

Solution: fxfg )((i)

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

f

x

1)(xg 22

x

1

)()()( fxffxff(iii) 22 x 2)( 2 22 x

212

x

)()( xggxgg(iv)

g

xg

1)( e.g. 1 Given that and find x

x64 24 xx

2

12

x

Functions

gxgf )((ii) )(xf )(g 22 x

Solution: fxfg )((i)

)()()()( xggxffxgfxfg (iv)(iii)(ii)(i)

2)( 2 xfx

g1

)( e.g. 1 Given that and find x x

f

x

1)(xg 22

x

1

)()()( fxffxff(iii) 22 x 2)( 2 22 x

212

x

)()( xggxgg(iv)

g

xg

1)( e.g. 1 Given that and find

x

1 1

x1

x64 24 xx

2

12

x

Functions

SUMMARY• A compound function is a function of a

function.

• It can be written as which means)(xfg .)(xgf

• is not usually the same as )(xgf .)(xfg

• The inner function is .)(xg

• is read as “f of g of x”. )(xgf

FunctionsExercise

,1)( 2 xxf

1. The functions f and g are defined as follows:

(a) The range of f is

Solution:

x 0x,

1)(

xxg

(a) What is the range of f ?

(b) Find (i) and (ii))(xfg )(xgf

1y

xf1

11

2

x

12xg1

12 x

(b) (i)

)(xgf)(xfg

(ii) )(xfg)(xgf

112

x

Functions

Periodic FunctionsFunctions whose graphs have sections which repeat are called periodic functions.

e.g.xy cos

This has a period of 3.

repeats every radians.

xcos2

It has a period of 2

Functions

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Functions

Some functions are even

Even functions are symmetrical about the y - axis

e.g. 2)( xxf

xxf cos)(

So, )()( xfxf e.g.

)2()2( ff

)()( ff

e.g.

Functions

Others are odd

Odd functions have 180 rotational symmetry about the origin

e.g.3)( xxf

xxf sin)(

)()( xfxf e.g.

)2()2( ff

e.g.

22 ff

Functions

Many functions are neither even nor odd e.g.

xxxf 2)( 2

Try to sketch one even function, one odd and one that is neither. Ask your partner to check.

Functions