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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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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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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
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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
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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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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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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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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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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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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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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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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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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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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
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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
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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
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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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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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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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-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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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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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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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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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
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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
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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
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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=
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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
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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
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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
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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
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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?
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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
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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
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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.
Its inverse will be a funcon.
d
e
f
Its inverse will be a funcon.
Its inverse will not be a funcon.
Its inverse will not be a funcon.
Its inverse will not be a funcon.
c
The inverse is a funcon
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
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Thinking More: Thinking More:
5. 6.d
e
f
The inverse is a funcon
The inverse is a funcon
The inverse is not a funcon
The graph of the inverse is the same as the
graph of the funcon.
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