17739325.AUS K Functions AUS

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FunctionsFunctions

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Mathletics 100% © 3P Learning SERIES TOPIC

K 114

If 5 f x x2= +^ h , then what is f 2-^ h?

What values are allowed for x in the equaon x= ?

What do we mean when we write f x1- ^ h?

If 5 f x x2= +^ h , then what is f 2-^ h?

What values are allowed for x in the equaon x= ?

What do we mean when we write f x1- ^ h?

FUNCTIONS

Answer these quesons, before working through the chapter.

Answer these quesons, after working through the chapter.

Funcons relate two variables together using the equals sign. They are used in every scienc

eld since they take in an input, and produce an output or result. It is important to know when a

relaonship is or is not a funcon.

But now I think:

What do I know now that I didn’t know before?

I used to think:



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K 142

Basics

Definition of Functions

You have used funcons before, you just haven’t known yet. Here are some examples of funcons:

• 2 y x=• 3 4 y x x

2= + +

• 3 yx

=

A funcon assigns each input value to a single output value – using a relaonship between the variables.

Here is an explanaon using the above examples:

In the above funcons, the input value has been inserted into the funcon as the x-value. The y-value is the output.

We say that a funcon ‘maps’ input values to output values.

This is the important part:

For a relaon to be a funcon, each input value can only map to one output. If any input value maps

to more than one output value, then the relaonship is NOT a funcon but is only a relaon.

Funcon: y x x3 42

= + +

Funcon: y x2=

Funcon: y 3 x

=

Input Output

Output

Output

Input

Input

-5 -10

14

27

2

3

Each input has one output

Each output comes from one input

This is a funcon

Each input value has one output value

An output can have more than one input

This is a funcon

An input value has more than one output

This is NOT a funcon

This is a relaon

Input

values

Input

values

Input

values

Output

values

Output

values

Output

values



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K 314

Basics

Function Notation, f x ^ h

In mathemacs there are special methods to write funcons. These all mean the same thing:

y =

y x2=

f x =^ h

f x x2=^ h

: f x

: f x x2

This is the notaon

used up unl now

The most

common notaon

Mapping notaon

They all dene a funcon. For example:

all dene a funcon that maps x to 2 x.

The notaon f x^ h is the most commonly used, here is an example:

If f x x x 32

= +^ h and g x x 2 1= -^ h then calculate the following

Somemes algebraic expressions are substuted into funcons.

a

c

a b

b

d

f 2^ h

f g0 0-^ ^h h

g 1^ h

f g2 1 3 1+ -^ ^h h

f 2 2 3 2

10

2= +

=

^ ^ ^h h h

1

0 3 0 2 0 1

0 1

2= + - -

= - -

=

^ ^ ^h h h6 6

6 6

@ @

@ @

2 1g 1 1

1

= -

=

^ ^h h

2 1 3 1 3 1

2 4 3 3

1

2= + + -

= + -

= -

2 1-^ ^ ^h h h6 6

6 6

@ @

@ @

3 2

f p

p

p

p

1

1

3 3 2

3 1

+

= + -

= + -

= +

^

^

h

h

g t

t

t

3

3

2

2 2

4

= +

= +

^

^

h

h

Substute 2 for every x

Substute 0 for every x into f x^ h and g x^ h Substute 1 for every x in f x^ h and -1 for every x in g x^ h

Substute 1 for every x

Let 3 2 f x x = -^ h and g x x 32

= +^ h . Find the following



FunctionsFunctions

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K 144

Basics

Vertical Line Test

The Vercal Line Test is a quick way to test whether or not a graph represents a funcon.

• A vercal line is moved through the graph. If it cuts the graph more than once then the graph is NOT a funcon.• If a vercal line is passed over the curve of a funcon, it will never cut the graph more than once.

Determine whether these relaons are funcons: 2 3 y x x 2

= - - and 9 x y2 2

+ =

2 3 y x x2

= - - 9 x y2 2

+ =

No vercal line cuts the curve more than once

` the relaon 2 3 y x x2

= - - is a funcon.

A vercal line can cut the curve more than once

` the relaon 9 x y2 2

+ = is NOT a funcon.

This is true because if a vercal line can cut the graph more than once, it means that there is more than one

y-value (output) for a specic x-value (input). Here are some general rules you can use:

All oblique lines are funcons All horizontal lines are funcons All parabolas are funcons All vercal lines are NOT funcons

(the vercal line cuts it innity

mes)

All polynomials are funcons All hyperbolas are funcons All exponenals are funcons All circles are NOT funcons

(the vercal line cuts it more

than once)

y

x

y

x

y

x

-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4

4

3

2

1

-1

-2

-3

-4

4

3

2

1

-1

-2

-3

-4

y y

x x

y

x

y

x

y

x

y

x

y

x



FunctionsFunctions

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K 514

Questions Basics

1. What is the dierence between a funcon and a relaon?

2. Idenfy if these are funcons or not.

a

a

c

c

b

b

d

d

2

3

1

2

-1

4

4

3

-2

-4

-1

5

-1

3

5

0

10

-8

0

2

2

6

3

9

3. Let’s say f x x 3 4= -^ h :

nd f 4^ h nd f 4-^ h

nd f n^ h nd f t 2^ h



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K 146

BasicsQuestions

4. Let’s say 4 f x x x 2= +^ h and 7 3h x x = -^ h :

a

c

e

g

b

d

f

h

nd f 2 9^ h.

nd f h1 2- -^ ^h h.

nd f h3 1 4 2- -^ ^h h.

nd the value of x if h x 31=-^ h . nd the value of x if f x2 6= -^ h .

nd 3h 2- ^ h.

nd h f 1 2 5- +^ ^h h.

nd f m h m2+^ ^h h.



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K 714

Questions Basics

a

c

b

d

e f

5. Use the vercal line test to determine which of these are funcons:

y

x

y

x

y

x

y

x

y

x

y

x



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K 148

Knowing More

Set notaon is a special mathemacal way for wring a set of numbers.

In set notaon, brackets are used and dierent types of brackets mean dierent things:

• ‘(‘ and ')' are used to write a set where the boundaries are excluded.

• ‘[‘ and ‘]’ are used to write a set where the boundaries are included.

The symbol 3 means ‘innity’ and ‘ 3- ' means ‘negave innity’.

The best way to understand set notaon is to use examples:

Write these in set notaon:

a

c

e

g

b

d

f

h

2 3 x1 #-

x 0$

or4 7 x x1 $

or3 5 6 8 x x1 1# #-

2 3 x 1#-

x 02

or3 0 x x 2# -

or10 5 1 8 x x1 1# #- -

, x 2 3! -^ @

, x 0 3! h6

, , x 4 7,3 3! -^ h h6

, , x 3 5 6 8,! -^ h6@

2,3 x ! - h6

, x 0 3! ^ h

, 3 , x 0,3 3! - -^ ^ h@

, , x 10 5 1 8,! - - ^ h6 @

Not including

Excluding

Excluding

Excluding Excluding

Including Including

Including

Including Innity can never be included

Including

Excluding

Including

The symbol ! means “is in the set”. So “ x ! ” means “ x is in the set …”

Set Notation



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K 914

Questions Knowing More

1. Answer these quesons:

a

a

c

e

g

i

b

d

f

h

j

b

c

What is the dierence between wring 3 x 1 and 3 x # ?

What do the symbols 3 and 3- mean?

What is the dierence between wring , x 2 2! -^ h and 2,2 x ! -6 @?

2. Write these inequalies in set notaon.

x1 51 1

x2 81 #-

x 22

or1 3 x x 2# -

or5 1 8 x x31 # # #- -

1 5 x# #

x2 81#-

x 2#

or1 3 x x1 $

or4 7 x x31 2#-



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K 1410

Knowing MoreQuestions

3. Write these sets as inequalies.

a

c

e

g

i

k

b

d

f

h

j

l

, x 1 6! ^ h

, x 1 6! ^ @

, x 5 10! -^ @

, x 73! -^ h

, , x 3 4 5 8,! -^ h6@ , , x 2 6 10 20,! ^h6 @

, , x 4 8,3 3! -^ h h6

, x 1 6! h6

, x 1 6! 6 @

, x 3 3! -^ h

, x 73! -^ @

, , x 0 6 7, 3! ^ ^ h@



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K 1114

Knowing More

Domain

All funcons have a domain. Somemes certain input values ( x-values) are not allowed in funcons. For example:

• In g x x=^ h , only posive values of x (including 0) are allowed. There is no real square root of anegave number.

• In f x x1

=^ h , x can not be 0. f 0^ h is undened. All other x-values are allowed.

The set of x-values which are allowed is called the domain.

• In g x x=^ h the domain is all numbers greater than or equal to 0.

• In f x x1

=^ h , the domain is all real numbers except 0.

There are two ways to write the domain: Using inequalies and using brackets (set notaon).

Write the domain of these funcons using inequalies, and then using set notaon

a bg x x=^ h f x x1

=^ h

x can only be posive or 0 x can be any numbe except 0

Using inequalies

Using Set Notaon Using Set Notaon

Using inequalies

x 0$

Greater than or equal to Strictly less than

or0 0 x x3 31 1 1 1-

, x 0 3! h6 , , x 0 0,3 3! -^ ^h h

To nd the domain, always think which values for x are permissible in the funcon.

Here are the graphs of g x^ h and f x^ h above:

Only the posive x-values (and 0) have y-values Each x-value has a y-value except 0 x =

-1 0 1 2 3 4 5 6 7 8 9

4

3

2

1

-1

-2

-3

-4

y y

x-5 -4 -3 -2 -1 0 1 2 3 4 5

x

4

3

2

1

-1

-2

-3

-4



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K 1412


4. Idenfy the value(s) for x (if any) that would make these funcons undened:

a

c

e

g

i

k

b

d

f

h

j

l

f x x1

=^ h

a x

x

12

=^ h

h x x=^ h

t x x 3= +^ h

r x

x x1 1

1=

- +^

^ ^h

h hd x

x x 3

1=

-^

^h

h

p x x= -^ h

g x x 2

1=

-^ h

b x 5 x

=^ h

f x x 4= -^ h

m x x2 1= -^ h

q x x 4

12

=-

^ h

Hint: Which values for x make

the denominator zero?

Hint: Which values for x

expression negave?

Hint: factorise rstChallenge

Quesons



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K 1314


a

a

c

c

e

e

b

b

d

d

f

f

u x

x 3

1=

+^ h

f x x=^ h

s x

x x2 7

1=

+ -^

^ ^h

h h

b x x

1=^ h

g x x1= -^ h

z x

x x7 12

12

=+ +

^ h

m x x 1= -^ h

g x x= -^ h

f x x x 20

12

=- -

^ h

m x

x x5 8

1=

- +^

^ ^h

h h

d x x 3

1=

+^ h

h x x3 2= +^ h

5. Write the domains for these funcon using inequalies:

6. Write the domains for these funcons using set notaon.



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K 1414

Knowing More

All funcons have a range. The range is the set of output values ( y-values or funcon values).

Range

Here are some examples:

Find the range of these funcons from their graphs:

a

c

b

d

f x x 42

= -^ h

g x x=^ h

a x x 42

= +^ h

h x x2 1= +^ h

f x^ h only has y-values greater than or equal to -4

` the range is: 4 y $-

This can also be wrien: , y 4 3! - h6

g x^ h only has posive y-values (including 0)

` the range is: y 0$

This can also be wrien: , y 0 3! h6

a x^ h only has y-values greater than or equal to 4

` the range is: 4 y $

This can also be wrien: , y 4 3! h6

h x^ h has all y-values.

` the range is all real numbers

This can also be wrien: , y 3 3! -^ h

y y

-4 -3 -2 -1 0 1 2 3 4

x

x

4

3

2

1

-1

-2

-3

-4-4 -3 -2 -1 0 1 2 3 4

8

7

6

5

4

3

2

1

-1 0 1 2 3 4 5 6 7

4

3

2

1

-1

-2

-3

-4

y

x

y

-4 -3 -2 -1 0 1 2 3 4

x

4

3

2

1

-1

-2

-3

-4



FunctionsFunctions

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K 1514


y

-4 -3 -2 -1 0 1 2 3 4

x

4

3

2

1

-1

-2

-3

-4

y

-4 -3 -2 -1 0 1 2 3 4

x

1

-1

-2

-3

-4

-5

-6

-7

y

-4 -3 -2 -1 0 1 2 3 4

x

4

3

2

1

-1

-2

-3

-4

y

-4 -3 -2 -1 0 1 2 3 4

x

4

3

2

1

-1

-2

-3

-4

7. Find the domain and range for these funcons from their graphs.

a

c

b

d

f x x1 3= -^ h

h x x 22= - -^ h

g x x x4 32

= - +^ h

sins x x=^ ^h h



FunctionsFunctions

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K 1416


y

-4 -3 -2 -1 0 1 2 3 4

x

4

3

2

1

-1

-2

-3

-4

y y

-3 -2 -1 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 4 5

x x

7

6

5

4

3

2

1

-1

7

6

5

4

3

2

1

-1

e

g

f

h

f x x2 2= -^ h

m x x 2= +^ h

a x x 2= +^ h

b x 4 x

=^ h

-1 0 1 2 3 4 5 6 7

4

3

2

1

-1

-2

-3

-4

y

x



FunctionsFunctions

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K 1714


8. Even though some relaons aren’t funcons, they sll have domain and range.

Answer the quesons about this relaon:

y

-4 -3 -2 -1 0 1 2 3 4

x

4

3

2

1

-1

-2

-3

-4

a

b

c

a

b

c

d

e

What is the equaon of this relaon?

What are the maximum and minimum values for y?

What is the range of this relaon?

What are the maximum and minimum values for x?

What is the domain of this relaon?

9. Somemes a funcon is only dened on a certain interval.

y

-4 -3 -2 -1 0 1 2 3 4

x

4

3

2

1

-1

-2

-3

-4

-5

What are the highest and lowest points for this funcon?

Find the domain of this funcon.

Find the range of this funcon.



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K 1418

Using Our Knowledge

Shifting Graphs Vertically, f x c!^ h

The graph of a funcon f x^ h can be used to draw graphs of f x c+^ h or f x c-^ h .

• The graph of f x c+^ h is simply the graph of f x^ h shied up c units.

• The graph of f x c-^ h is simply the graph of f x^ h shied down c units.

Here is an example using a polynomial.

This is the graph of a funcon f x ^ h

a bDraw the graph for 3 f x +^ h . Draw the graph for 2 f x -^ h .

y

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

x

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

y

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

x

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

y

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

x

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

f x^ h

f x^ h f x 3+^ hup units3

down units2



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K 1914

Using Our Knowledge

Shifting Graphs Horizontally, f x c!^ h

The graph of a funcon f x^ h can be used to draw graphs of f x c+^ h or f x c-^ h.

• The graph of f x c+^ h is simply the graph of f x^ h shied le c units.

• The graph of f x c-^ h is simply the graph of f x^ h shied right c units.

Here is an example using a polynomial.

This is the graph of a funcon f x x x 42

= - -^ h

a bDraw the graph for f x 3-^ h. Draw the graph for 4g x x x1 12

= - + - +^ ^ ^h h h.

y

x

5

4

3

2

1

-1

-2

-3

-4

-5

-5 -4 -3 -2 -1 0 1 2 3 4 5

y y

x x

5

4

3

2

1

-1

-2

-3

-4

-5

5

4

3

2

1

-1

-2

-3

-4

-5

-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5

` shi f x^ h 3 units to the right This is simply f x 1+^ h.

` shi f x^ h 1 unit to the le

f x^ h

f x^ h f x^ h f x 3-^ h

f x1+

^ h

right units3left unit1



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K 1420

Using Our Knowledge

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

y

x

Inside and Outside the Brackets

When shiing graphs, you don’t even need to know what the original funcon is. Just remember that:

• If the constant is outside the bracket like f x c!^ h then the graph shis vercally.

• If the constant is inside the bracket like f x c!^ h then the graph shis horizontally.

• f x^ h is shied 2 units to the right

• f x^ h is shied 3 units upwards

Outside the bracket

Inside the bracket

Here is an example which has both.

The graph below represents f x ^ h. Use it to draw the graph of 3 f x 2- +^ h

The new graph 3 f x 2- +^ h has -2 inside the bracket and +3 outside the bracket. This means that:

y

x-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

Step 1

: Shi the graph2

units to the right Step 2

: Shi this graph3

units upwards Final graph:3

f x2- +

^ h y

x-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

y

x-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

units2units3



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K 2114

Questions Using Our Knowledge

1. The graph below is of 2 1 f x x = +^ h . Draw these graphs on the same set of axes:

a

a

b

b

c

d

f x 2+^ h

f x 2+^ h

f x 3-^ h

f x 3-^ h

2. The graph below represents g x ^ h. Use it to draw g x 4+^ h on the other set of axes.

3. Explain the dierence between the following graphs, if f x ^ h is any funcon.

f x 2+^ h and f x 2-^ h

f x 4+^ h and f x 4+^ h

-4 -3 -2 -1 0 1 2 3 4

4

3

2

1

-1

-2

-3

-4

y

x

y

x-5 -4 -3 -2 -1 0 1 2 3 4 5 6

5

4

3

2

1

-1

-2

-3

-4

-5

y

x-5 -4 -3 -2 -1 0 1 2 3 4 5 6

5

4

3

2

1

-1

-2

-3

-4

-5

f x^ h



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K 1422

Using Our KnowledgeQuestions

4. The graph below is of 2 5 6 f x x x x 3 2= - - + +^ h . Draw these graphs on the same set of axes:

a

b

c

f x 4+^ h

f x 4-^ h

g x x x x2 2 2 5 2 63 2

= - - - - + - +^ ^ ^ ^h h h h

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

10

9

8

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

y

x



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K 2314

Questions Using Our Knowledge

y

x-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

5. The graph below is of f x ^ h. Use it to draw 3 f x 1+ -^ h on the other set of axes:

6. The graph on the right shows the funcon 7 f x 5+ +^ h . Draw the original f x ^ h on the le set of axes.

-4 -3 -2 -1 0 1 2 3 4

4

3

2

1

-1

-2

-3

-4

y

x-4 -3 -2 -1 0 1 2 3 4

4

3

2

1

-1

-2

-3

-4

y

x

y

x-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

f x^ h

f x 5 7+ +^ h



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K 1424

Thinking More

Inverses, ( ) x f 1-

2 y x= and yx

2= have the inverse eect on any number. Each one is the inverse of the other.

An inverse funcon reverses the eect of the original funcon.

For example:

f x^ h

x2

x2

f x^ h

10

Inverse of f x^ h1st

output

1st

output

2nd

intput

2nd

intput

2nd

output

2nd

output

1st

input

1st

input

x

5

x

5

Same as 1st

input

The inverse of f x^ h is wrien as f x1- ^ h.

The easiest way to nd an inverse is to switch the pronumerals and then solve for y.

Find the inverse of these funcons:

a

c

f x x

y x

2

2`

=

=

^ h

f x x

y x

1

1

3

3`

= -

= -

^ h

g x x

y x

3 1

3 1`

= -

= -

^ h

p x x

y x

4

4`

=

=

^ h

y x

f x x

2

2

1

`

`

=

=- ^ h

y x

f x x

1

1

3

1 3

`

`

= +

= +- ^ h

yx

g xx

3

1

3

11

`

`

=+

=+- ^ h

y x

p x x

4

41

`

`

=

=

-

^ h

For inverse: x y2=

For inverse: x y 13

= -

For inverse: x y3 1= -

For inverse: x y

4=

Switch x and y

Solve for y

Replace f x^ h with y

b

d

This means "inverse", not f x

1

^ h



FunctionsFunctions

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K 2514

Questions Thinking More

1. Answer these quesons about these funcons:

5 10 f x x = -^ h

g x x

4 2

1

= +^ h

h x x 4 2= -^ h

m x x

52= +^ h

(i)

(ii)

(iii)

(iv)

a d

b e

c f

Find f 2^ h. Find g 1-^ h.

Substute this value into h x^ h and m x^ h. Substute this value into h x^ h and m x^ h.

Is h x^ h or m x^ h the inverse of f x^ h. Is h x^ h or m x^ h the inverse of g x^ h.

2. Match each funcon to its inverse.

1

2

6

3

7

4

8

5

9

f x x3 2= +^ h

f x x10 5= -^ h

f x x 32

= -^ h

f x x6 4= - +^ h

f x x25

1= -^ h

f x x3

2= +

^ h

f x x 23

= +^ h

f x x 23

= -^ h

f x x6 4= +^ h

f x x6 3

21= - +

- ^ ha

f x x 31

= -- ^ hb

f x x3

21=

-- ^ hc

f x

x

2 10

11= +

-

^ hd

f x x 21 3

= +- ^ he

f x x6 3

21= -

- ^ hf

f x x 31

= +- ^ hg

f x x10 2

11= +

- ^ hh

f x x 21 3

= -- ^ hi



FunctionsFunctions

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K 1426

Thinking MoreQuestions

3. Find the inverse of each of these funcons:

4a x x=^ h

c xx

7

3 2=

+^ h

g x x 13=+^ h

e x x 13

= +^ h

b x x2 4= -^ h

d x x5

4

2

3= -^ h

h x mx c= +^ h

f x x3

=^ h

a

c

g

e

b

d

h

f



FunctionsFunctions

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K 2714

Thinking More

y

x-3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

y

x-3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

Inverse Graphs and Inverse Functions

Here are the graphs of 2 3 f x x= -^ h and f x x2

31=

+- ^ h on the same set of axes:

As you can see, the graph of f x1- ^ h is simply the reecon of f x^ h around the line y x= . This makes sense since

the inverse was found by switching x and y in the equaon.

This is always the case.

To draw any inverse f x1- ^ h, simply nd the reecon of f x^ h around the line y x= . Here is another example:

f x1- ^ h will always intersect with f x^ h over the line y x= .

The graph below is of 4 3 f x x x 2= - +^ h . Find the graph of f x 1- ^ h:

y

x-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

f x x- ^ h

f x^ h y x=

Flip f x^ h

around y x=



FunctionsFunctions

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K 1428

Thinking More

Using the vercal line test, it’s easy to see that the inverse of a parabola is not a funcon, but the inverse of a straight

line is a funcon. This means that an inverse isn’t always an inverse funcon. When do inverse funcons exist?

Remember a funcon is a relaon which has only one output value for each input value.

Funcons can be divided into two main types:

• Many-to-one funcons: Although each input value must only have

one output value, the same output value could come from more

than one (many) input values.

• One-to-one funcons: Each output value comes from a dierent

input value.

f x1- ^ h will only be a funcon if it is one-to-one. If so, then f x1- ^ h will also be one-to-one.

A simple test to determine whether or not a funcon is one-to-one or many-to-one is the horizontal line test.

• If any horizontal line does cut the graph more than once then the graph is a many-to-one funcon.

• If no horizontal line can cut the graph more than once then the graph is a one-to-one funcon.

Input values

Input values

Output values

Output values

This funcon is one-to-one since no horizontal

line cuts the graph twice

f x 1`

- ^ h will be a funcon. g x 1

`- ^ h will not be a funcon.

This funcon is many-to-one since there is a

horizontal that cuts the graph more than once.

Test whether f x ^ h and g x ^ h below will have an inverse funcons.

y y

x x-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

5

4

3

2

1

-1

-2

-3

-4

-5



FunctionsFunctions

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K 2914

Thinking More

As usual, to nd the inverses of f x^ h and g x^ h on the previous page, just reect the graph around the line y x= :

The vercal line test can be used on the graphs of f x1- ^ h and g x1- ^ h to test whether or not these are funcons.

1. A relaon is an expression involving two variables ( x and y ) and the equals (=) sign.

2. Funcons are relaons that have only one output for every input.

3. In funcons, it is possible for many inputs to have the same output. It is impossible to have many outputs

for a single input.

4. Funcons can be given names like f x^ h or g x^ h. This is called funcon notaon.

5. The vercal line test is used (on a graph) to test whether or not a relaon is a funcon.

6. The domain of a funcon is the set of allowed x-values (input values).

7. The range of a funcon is the set of y-values (output values).

8. The graph of f x c!^ h is just the graph of f x^ h shied up (+) or down (-).

9. The graph of f x c!^ h is just the graph of f x^ h shied le (+) or right (-).

10. The inverse of a funcon f x^ h is wrien as f x1- ^ h

11. f x1- ^ h will only be a funcon if f x^ h is a one-to-one funcon.

12. f x f x1 1=

- -^ ^ ^h h h. The inverse, of an inverse is the original funcon.

Here are some important points to remember about funcons:

y

x-4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

y

x-4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5



FunctionsFunctions

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K 1430


4. Idenfy if these graphs represent a one-to-one or many-to-one funcon and state whether or not its

inverse is a funcon.

a

c

e

b

d

f

y

x

y

x

y

x

y

x

y

x

y

x



FunctionsFunctions

100% Functons


K 3114

Questions Thinking More

5. Sketch the inverse of these graphs on the same axes. State whether or not the inverse is a funcon.

a

c

b

d

y

x-4 -3 -2 -1 0 1 2 3 4

4

3

2

1

-1

-2

-3

-4

y

x-4 -3 -2 -1 0 1 2 3 4

4

3

2

1

-1

-2

-3

-4

y

y

x

x

-4 -3 -2 -1 0 1 2 3 4

-4 -3 -2 -1 0 1 2 3 4

4

3

2

1

-1

-2

-3

-4

4

3

2

1

-1

-2

-3

-4



FunctionsFunctions

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K 1432


y

x-4 -3 -2 -1 0 1 2 3 4

4

3

2

1

-1

-2

-3

-4

e f y

x-4 -3 -2 -1 0 1 2 3 4

4

3

2

1

-1

-2

-3

-4

y

x-4 -3 -2 -1 0 1 2 3 4

4

3

2

1

-1

-2

-3

-4

6. Use the graph of f x x 3

=^ h below to draw f x 1- ^ h. What do you noce?



Functions

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Mathletics 100% © 3P Learning

Functions

SERIES TOPIC

K 1434

Answers

x can be any number except -1 or 1

x can be any number except 0 or 3.

x can be any number except -2 or 2

x can only be negave or 0.

Knowing More: Knowing More:

Using Our Knowledge:

i

j

k

l

4.

5.

6.

7.

7.

8.

1.

9.

a

a

a

c

c

c

e

e

e

g

b

b

b

d

d

d

f

f

or3 3 x x3 31 1 1 1- - -

or

or

x x

x

2

2 7

7

3

3

1 1

1 1

1 1

- -

-

or

or

x

x

x

4

4 5

5

3

3

1 1

1 1

1 1

- -

-

x 1$

x 1#

x 32-

, x 0 3! h6

, x 0 3! ^ h

, , , x 8 8 5 5, ,3 3! - - -^ ^ ^h h h

, , , x 4 4 3 3, ,3 3! - - - - -^ ^ ^h h h

, x 03! -^ @

, x32

3! - j8

,

,

x

y

3 3

3 3

!

!

-

-

^

^

h

h

,

1,

x

y

3 3

3

!

!

-

-

^ h

h6

,

, 2

x

y

3 3

3

!

!

-

- -

^

^

h

@

,

1,1

x

y

3 3!

!

-

-

^ h

6 @

f

h

,

,

x

y 0

3 3

3

!

!

-^ h

h6

,

,

x

y

0

2

3

3

!

!

h

h

6

6

,

,

x

y 0

3 3

3

!

!

-^

^

h

h

,

,

x

y

2

0

3

3

!

!

- h

h

6

6

a

b

c

d

e

Maximum value of y is 3

Minimum value of y is-3

y3 3# #-

3 3 x# #-

Maximum value of x is 3

Minimum value of x is -3

a

b

c

Highest point is at ,1 4^ hLowest points are at ,2 5- -^ h and ,4 5-^ h

x2 4# #-

4 y5 # #-

9 x y2 2

+ =

a b d c



Functions

100% Functons


Functions

SERIES TOPIC

K 3514

Answers

Using Our Knowledge: Using Our Knowledge:

Thinking More:

2. 5.

6.

1.

3.

4.

a

b

f x 2+^ h is f x^ h shied two units to

the le, whereas f x 2-^ h is f x^ h

shied two units to the right.

f x 4+^ h is f x^ h shied 4 units to the

le, whereas 4 f x +^ h is f x^ h shied

4 units upwards.

f x^ h is shied 4 units to the le

f x^ h is shied 4 units downwards

The graph of g x^ h is the graph of f x^ h shied 2 units to the right

a

a

b

c

b

c

a

b

c

f 2 0=^ h

h 0 2= -^ h

m 0 2=^ h

m x^ h is the inverse of f x^ h

3 f x 1+ -^ h

f x^ h



Functions

100% Functons


Functions

SERIES TOPIC

K 1436

Answers

g 14

1- =^ h

1h41 =-` j m

41

2041=` j

h x^ h is the inverse of g x^ h

Thinking More: Thinking More:

1. 5.

2.

3.

4.

d

e

f

1

2

3

4

5

f x x3 2= +^ h

f x x10 5= -^ h

f x x6 4= - +^ h

f x x 32

= +^ h

f x x 23

= -^ h

a

a

b

a

ba xx

4

1=

- ^ h b xx

4

21=

-

-- ^ h

c

c

dc xx

3

7 21=

-- ^ h d xx

4

5

8

151= +

- ^ h

g

e

h

f e x x 11 3

= -- ^ ^h h

g x x

31

1= -

-

^ h

f x x31

=- ^ h

h xm

x c1=

--

^ h

b

Its inverse will be a funcon.


d

e

f


Its inverse will not be a funcon.



c

The inverse is a funcon


The inverse is not a funcon

6

7

8

9

f x x 32

= -^ h

f x x25

1= -^ h

f x x 23

= +^ h

f x x6 4= +^ h

f x x6 3

21= - +

- ^ ha

f x x 31

= -- ^ hb

e

f x x3

21=

-- ^ hc

f x x10 2

11= +

- ^ hh

f x x 21 3

= +- ^ h

f x x2 10

11= +

- ^ hd

f xx6 3

21

= -

-

^ hf

f x x 31

= +- ^ hg

f x x 21 3

= -- ^ hi



Functions

100% Functons


Functions

SERIES TOPIC

K 3714

Answers

Thinking More: Thinking More:

5. 6.d

e

f



The inverse is not a funcon

The graph of the inverse is the same as the

graph of the funcon.



FunctionsFunctions

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K 1438

Notes



FunctionsFunctions

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K 3914

Notes



FunctionsFunctions

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K 1440

Notes

17739325.AUS K Functions AUS

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