Functions Part IRecap

)(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.

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.

so the range is 5y

The graph of looks like 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 )Rx

3 xy

domain: x-values range: y-values3x 0y

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

3)( xxf)(xfy )(xf

The graph looks like:

( We could write instead of y ) )( xf

1 xy

Is a function ? 1 xyA function must have only one y value for each x value

In other words each X gives only one F(x)

Notation for Domain/Range

Rx X exists for all Real numbers “R”

Counting numbers Z+ Positive whole numbers: 1, 2, 3 ….Natural numbers N Counting numbers and zero: 0, 1, 2, 3 Integers Z All whole numbers: ..-2, -1, 0, 1, 2, ..Rational numbers Q m/n where m and n are integers n≠0.. -1, 0, ½,1, 2¾ Real numbers R Rational and irrational* number .. -1, 2¾ , π, √2

Part IIInverse functions

What’ the inverse of f(x)=2x+4 ?

X F(X)

X F(X)

The inverse function f-1(x) is the reverse of f(x)

Calculating the Inverse

In practice this is how we calculate the inverse

Example: f(x)=2x+4

42 xy

Consider the graph of the function

42)( xxf

The inverse function is 2

4)(1 x

xf

42 xy

2

4

xy

42 xy

2

4

xy

Consider the graph of the function

42)( xxf

The inverse function is 2

4)(1 x

xf

An inverse function is just a rearrangement with x and y swapped.

So the graphs just swap x and y!

)4,0(x

)0,4(x

)2,3( x

)3,2( x

42 xy

2

4

xy

)4,0(x

)0,4(

x

)2,3( x

)3,2( x

is a reflection of in the line y = x

)(1 xf )( xf

xy

What else do you notice about the graphs?

)4,4( x

The function and its inverse must meet on y = x

e.g.On the same axes, sketch the graph of

and its inverse.

2,)2( 2 xxy

N.B!

)0,2(

)1,3(

xy

)4,4(x

Solution:

)2,0(

)3,1(

e.g.On the same axes, sketch the graph of

and its inverse.

2,)2( 2 xxy

N.B!xy

2)2( xy

Solution:

2 xy

2)2( xy

2 xy

A bit more on domain and range

The domain of is . 2x

)( xf

Since is found by swapping x and y,

)(1 xf

2)2()( xxf 2xDomain

2y2)(1 xxf Range

2,)2()( 2 xxxfThe previous example used .

the values of the domain of give the values of the range of .

)( xf)(1 xf

2)2( xy

A bit more on domain and range

2,)2()( 2 xxxfThe previous example used .

The domain of is . 2x

)( xf2 xy

Since is found by swapping x and y,

)(1 xf

)(1 xf give the values of the domain of

the values of the domain of give the values of the range of .

)( xf)(1 xf

Similarly, the values of the range of

)(xf

1 xy

N.B. The curve below does not have an inverse because it is not a function

We can get around this

By splitting it into two parts

( 2 y’s for 1 x)

Part IIIFunction of a function

then,

)(f 3

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

x is replaced by 3

)( xgand f

2)( 3

Suppose and2)( xf )( xg

)(f 1

then,

)(f 32)( 1

91

x 3xFunctions of a Function

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

3x)( xgand f

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

f )( xg

2)( 3x962 xx

f

2)( 3

Suppose and2)( xf )( xg

)(f 1

then,

)(f 32)( 1

91

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

Functions of a Function

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

Solution:

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

2)( 2 xfx

g1

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

fxfg )((i) )( xg

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

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

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

21

2

x

)( 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

21

2

x

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

21

2

x

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)

21

2

x

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

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)

21

2

x

2

12

x

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

21

2

x

2

12

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) 22 x 2)( 2 22 x

21

2

x

)()( xggxgg(iv)64 24 xx

2

12

x

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

21

2

x

)()( xggxgg(iv)

g

xg

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

x64 24 xx

2

12

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) 22 x 2)( 2 22 x

21

2

x

)()( xggxgg(iv)

g

xg

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

x

1 1

x1

x64 24 xx

2

12

x

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

xxffxff )()( 11

N.B. The order in which we find the compound function of a function and its inverse makes no difference.

For all functions which have an inverse,)( xf

Exercise

,1)( 2 xxf1. The functions f and g are defined as follows:

(a) The range of f is Solution:

x R

0x,1

)(x

xg

(a) What is the range of f ?

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

1y

xf

11

12

x

12xg1

12 x

(b) (i) )(xgf)( xfg

(ii) )(xfg)( xgf

11

2

x