Functions
Functions
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Functions
FUNCTIONS
Functions relate two variables together using the equals sign. They are used in every scientific
field since they take in an input, and produce an output or result. It is important to know when a
relationship is or is not a function.
Answer these questions, before working through the chapter.
I used to think:
If f ^ xh = x2 + 5 , then what is f ^-2h ?
What values are allowed for x in the equation =
x?
What do we mean when we write f -1 ^ xh ?
Answer these questions, after working through the chapter.
But now I think:
If f ^ xh = x2 + 5 , then what is f ^-2h ?
What values are allowed for x in the equation =
x?
What do we mean when we write f -1 ^ xh ?
What do I know now that I didn’t know before?
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Functions
Basics
Definition of Functions
You have used functions before, you just haven’t known yet. Here are some examples of functions:
•
•
•
y = 2x
y = x 2 + 3x + 4
y = 3x
A function assigns each input value to a single output value – using a relationship between the variables.
Here is an explanation using the above examples:
Input
-5
Input
2
Input
3
Function: y = 2x
Function: y = x2 + 3x + 4
Function: y = 3 x
Output
Output
Output
-10
14
27
In the above functions, the input value has been inserted into the function as the x-value. The y-value is the output.
We say that a function ‘maps’ input values to output values.
This is the important part:
For a relation to be a function, each input value can only map to one output. If any input value maps
to more than one output value, then the relationship is NOT a function but is only a relation.
Input
values
2
Output
values
Input
values
Output
values
Input
values
Output
values
Each input has one output
Each input value has one output value
An input value has more than one output
Each output comes from one input
An output can have more than one input
This is NOT a function
This is a function
This is a function
This is a relation
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Functions
Basics
Function Notation, f^ xh
In mathematics there are special methods to write functions. These all mean the same thing:
y=
This is the notation
used up until now
f ^ xh =
f :x
The most
common notation
Mapping notation
They all define a function. For example:
f ^ xh = 2x
y = 2x
f :x
2x
all define a function that maps x to 2x.
The notation f ^ xh is the most commonly used, here is an example:
If f ^ xh = x 2 + 3x and g^ xh = 2x - 1 then calculate the following
a
f ^2h
b
Substitute 2 for every x
g^1 h
Substitute 1 for every x
f ^2h = ^2h2 + 3^2h
g^1 h = 2^1 h - 1
= 10
c
=1
f ^0h - g^0h
Substitute 0 for every x into
d
f ^ xh and g^ xh
2f ^1 h + 3g^-1h
Substitute 1 for every x in
f ^ xh and -1 for every x in g^ xh
= 26^1 h2 + 3^1 h@ + 3 62^-1h - 1 @
= 6^0h2 + 3^0h@ - 62^0h - 1 @
= 264 @ + 3 6-3 @
= 60 @ - 6-1 @
=-1
=1
Sometimes algebraic expressions are substituted into functions.
Let f ^ xh = 3x - 2 and g ^ xh = x 2 + 3 . Find the following
a
f ^ p + 1h
b
g ^t 2 h
= ^t 2 h + 3
= 3^ p + 1h - 2
2
= t4 + 3
= 3p + 3 - 2
= 3p + 1
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Functions
Basics
Vertical Line Test
The Vertical Line Test is a quick way to test whether or not a graph represents a function.
•
•
A vertical line is moved through the graph. If it cuts the graph more than once then the graph is NOT a function.
If a vertical line is passed over the curve of a function, it will never cut the graph more than once.
Determine whether these relations are functions: y = x 2 - 2x - 3 and x 2 + y 2 = 9
y = x2 - 2x - 3
x2 + y2 = 9
y
-4
-3
-2
y
4
4
3
3
2
2
1
1
-1 0
-1
1
2
3
x
4
-4
-3
-2
-1 0
-1
-2
-2
-3
-3
-4
-4
No vertical line cuts the curve more than once
` the relation y = x2 - 2x - 3 is a function.
1
2
3
4
x
A vertical line can cut the curve more than once
` the relation x2 + y2 = 9 is NOT a function.
This is true because if a vertical line can cut the graph more than once, it means that there is more than one
y-value (output) for a specific x-value (input). Here are some general rules you can use:
y
y
y
x
x
All horizontal lines are functions
All parabolas are functions
y
y
y
All polynomials are functions
4
x
All oblique lines are functions
x
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x
All hyperbolas are functions
x
All vertical lines are NOT functions
(the vertical line cuts it infinity
times)
y
x
All exponentials are functions
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x
All circles are NOT functions
(the vertical line cuts it more
than once)
Functions
Questions
Basics
1. What is the difference between a function and a relation?
2. Identify if these are functions or not.
a
b
2
1
5
3
3
-2
0
-8
-1
4
2
c
d
3
2
-1
9
-4
3
-1
10
4
6
3. Let’s say f ^ xh = 3 - 4x :
a
find f ^4h
b
find f ^-4h
c
find f ^ nh
d
find f ^2t h
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0
2
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Functions
Questions
4. Let’s say f^ xh = x 2 + 4x and h^ xh = 7x - 3 :
Basics
a
find 2f ^9h .
b
find -3h^2h .
c
find f ^1 h - h^-2h .
d
find h^-1h + 2f ^5h .
e
find 3f ^-1h - 4h^2h .
f
find f ^ mh + h^ m2h .
g
find the value of x if h^ xh =-31 .
h
find the value of x if 2f ^ xh =-6 .
6
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Functions
Questions
Basics
5. Use the vertical line test to determine which of these are functions:
a
b
y
y
x
c
x
d
y
y
x
e
x
f
y
y
x
x
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Knowing More
Set Notation
Set notation is a special mathematical way for writing a set of numbers.
In set notation, brackets are used and different types of brackets mean different 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 ‘infinity’ and ‘-3 ' means ‘negative infinity’.
The best way to understand set notation is to use examples:
Write these in set notation:
a
b
-2 1 x # 3
Not including
Including
-2 # x 1 3
Including
x ! 6-2, 3h
x ! ^-2, 3 @
c
x20
d
x$0
Including
Excluding
x ! ^0, 3h
x ! 60, 3h
Including
e
Infinity can never be included
x 1 4 or
Excluding
Excluding
x # -3 or
f
x$7
Including
Including
x ! ^-3, 4h , 67, 3h
g
-3 1 x # 5 or 6 # x 1 8
h
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Excluding
-10 # x # -5 or 1 1 x 1 8
x ! 6-10, -5 @ , ^1, 8h
The symbol ! means “is in the set”. So “ x !” means “x is in the set …”
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x20
x ! ^-3, -3 @ , ^0, 3h
x ! ^-3, 5 @ , 66, 8h
8
Excluding
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Functions
Questions
Knowing More
1. Answer these questions:
a
What is the difference between writing x 1 3 and x # 3 ?
b
What do the symbols 3 and -3 mean?
c
What is the difference between writing x ! ^-2, 2h and x ! 6-2, 2 @?
2. Write these inequalities in set notation.
a
11x15
b
1#x#5
c
-2 1 x # 8
d
-2 # x 1 8
e
x22
f
x#2
g
x # -1 or
h
x 1 1 or
i
-5 1 x # -1 or 3 # x # 8
j
-4 # x 1 3 or
x23
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x$3
x27
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Functions
Questions
3. Write these sets as inequalities.
a
x ! ^1, 6h
b
x ! 61, 6h
c
x ! ^1, 6 @
d
x ! 61, 6 @
e
x ! ^-5, 10 @
f
x ! ^-3, 3h
g
x ! ^-3, 7h
h
x ! ^-3, 7 @
i
x ! ^-3, 4 @ , 65, 8h
j
x ! 62, 6h , ^10, 20 @
k
x ! ^0, 6 @ , ^7, 3h
l
x ! ^-3, 4h , 68, 3h
10
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Knowing More
Functions
Knowing More
Domain
All functions have a domain. Sometimes certain input values (x-values) are not allowed in functions. For example:
In g^ xh = x , only positive values of x (including 0) are allowed. There is no real square root of a
negative number.
In f ^ xh = 1 , x can not be 0. f ^0h is undefined. All other x-values are allowed.
x
•
•
The set of x-values which are allowed is called the domain.
In g^ xh =
•
x the domain is all numbers greater than or equal to 0.
1
In f ^ xh = , the domain is all real numbers except 0.
x
•
There are two ways to write the domain: Using inequalities and using brackets (set notation).
Write the domain of these functions using inequalities, and then using set notation
a
g^ x h =
b
x
f ^ xh = 1
x
x can only be positive or 0
x can be any numbe except 0
Using inequalities
Using inequalities
-3 1 x 1 0 or 0 1 x 1 3
x$0
Strictly less than
Greater than or equal to
Using Set Notation
Using Set Notation
x ! 60, 3h
x ! ^-3, 0h , ^0, 3h
To find the domain, always think which values for x are permissible in the function.
Here are the graphs of g^ xh and f ^ xh above:
y
-1
y
4
4
3
3
2
2
1
1
0
1
2
3
4
5
6
7
8
9
x
-5
-4
-3
-2
-1
0
-1
-1
-2
-2
-3
-3
-4
-4
Only the positive x-values (and 0) have y-values
1
2
3
4
x
5
Each x-value has a y-value except x = 0
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Questions
Knowing More
4. Identify the value(s) for x (if any) that would make these functions undefined:
a
f ^ xh = 1
x
b
g^ x h =
c
a^ xh = 12
x
d
b^ x h = 5 x
e
h^ xh =
x
f
f ^ xh =
g
t ^ xh =
x+3
h
m^ xh = 2x - 1
j
d ^ xh =
l
p^ xh = -x
r ^ xh =
i
k
12
1
^ x - 1h^ x + 1h
q^ xh =
1
x2 - 4
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Hint: factorise first
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1
x-2
x-4
Hint: Which values for x make
the denominator zero?
Hint: Which values for x
expression negative?
1
x^ x - 3h
Challenge
Questions
Functions
Questions
Knowing More
5. Write the domains for these function using inequalities:
a
u^ x h =
c
s^ x h =
e
g^ x h = 1 - x
1
x+3
1
^ x + 2h^ x - 7h
b
m^ x h =
d
f ^ xh =
f
d ^ xh =
x-1
1
x2 - x - 20
1
x+3
6. Write the domains for these functions using set notation.
a
f ^ xh =
x
b
g^ xh = -x
c
b^ x h = 1
x
d
m^ x h =
e
z ^ xh =
f
h ^ x h = 3x + 2
1
x2 + 7x + 12
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1
^ x - 5h^ x + 8h
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Functions
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Range
All functions have a range. The range is the set of output values (y-values or function values).
Here are some examples:
Find the range of these functions from their graphs:
a
f ^ x h = x2 - 4
-4
-3
-2
b
y
-1
a ^ x h = x2 + 4
4
8
3
7
2
6
1
5
0
1
2
3
4
x
4
-1
3
-2
2
-3
1
-4
-4
f ^ xh only has y-values greater than or equal to -4
c
-3
-2
-1
0
1
2
3
` the range is: y $ 4
This can also be written: y ! 6-4, 3h
This can also be written: y ! 64, 3h
g^ x h =
y
d
x
h^ xh = 2x + 1
4
3
3
2
2
1
1
1
2
3
4
5
6
7
x
-4
-3
-2
-1
0
-1
-1
-2
-2
-3
-3
-4
-4
g^ xh only has positive y-values (including 0)
1
2
3
4
h^ xh has all y-values.
` the range is: y $ 0
` the range is all real numbers
This can also be written: y ! 60, 3h
This can also be written: y ! ^-3, 3h
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x
y
4
0
4
a^ xh only has y-values greater than or equal to 4
` the range is: y $ -4
-1
14
y
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x
Functions
Questions
Knowing More
7. Find the domain and range for these functions from their graphs.
a
f ^ x h = 1 - 3x
b
g^ xh = x2 - 4x + 3
y
-4
c
-3
-2
-1
y
4
4
3
3
2
2
1
1
0
1
2
3
4
x
-4
-3
-2
-1
0
-1
-1
-2
-2
-3
-3
-4
-4
h^ xh =-x2 - 2
d
-2
-1
0
3
4
1
2
3
4
x
y
1
-3
2
s^ xh = sin ^ xh
y
-4
1
4
1
2
3
4
x
3
-1
2
-2
1
-3
-4
-3
-2
-1
0
-4
-1
-5
-2
-6
-3
-7
-4
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15
Functions
e
Questions
f ^ xh = 2x - 2
a^ xh =
f
g
-3
m^ x h =
-2
-1
4
4
3
3
2
2
1
1
0
x +2
y
y
-4
Knowing More
1
2
3
4
x
0
-1
-1
-1
-2
-2
-3
-3
-4
-4
h
x+2
1
2
-2
-1
7
6
6
5
5
4
4
3
3
2
2
1
1
1
2
3
4
5
x
-3
-1
16
K 14
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TOPIC
5
6
7
2
3
4
5
x
y
7
0
4
b^ x h = 4 x
y
-3
3
-2
-1
0
-1
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1
x
Functions
Questions
Knowing More
8. Even though some relations aren’t functions, they still have domain and range.
Answer the questions about this relation:
a
y
What is the equation of this relation?
4
3
b
What are the maximum and minimum values for y?
2
1
c
-4
What is the range of this relation?
-3
-2
-1
0
1
2
3
4
1
2
3
4
x
-1
-2
-3
d
What are the maximum and minimum values for x?
e
What is the domain of this relation?
-4
9. Sometimes a function is only defined on a certain interval.
a
y
What are the highest and lowest points for this function?
4
3
2
1
b
Find the domain of this function.
-4
-3
-2
-1
0
x
-1
-2
-3
-4
c
-5
Find the range of this function.
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Functions
Using Our Knowledge
Shifting Graphs Vertically, f ^ xh ! c
The graph of a function f ^ xh can be used to draw graphs of f ^ xh + c or f ^ xh - c .
•
•
The graph of f ^ xh + c is simply the graph of f ^ xh shifted up c units.
The graph of f ^ xh - c is simply the graph of f ^ xh shifted down c units.
Here is an example using a polynomial.
This is the graph of a function f ^ xh
y
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0
-1
1
2
3
4
5
6
x
-2
-3
-4
f ^ xh
a
-5
-6
Draw the graph for f ^ xh + 3 .
b
Draw the graph for f ^ xh - 2 .
y
up 3 units
y
6
6
f ^ xh + 3
5
4
f ^ xh
5
4
3
3
2
2
1
-7 -6 -5 -4 -3 -2 -1 0
-1
18
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1
1
2
3
4
5
6
x
-7 -6 -5 -4 -3 -2 -1 0
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
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2
3
4
down 2 units
5
6
x
Functions
Using Our Knowledge
Shifting Graphs Horizontally, f ^ x ! ch
The graph of a function f ^ xh can be used to draw graphs of f ^ x + ch or f ^ x - ch .
•
•
The graph of f ^ x + ch is simply the graph of f ^ xh shifted left c units.
The graph of f ^ x - ch is simply the graph of f ^ xh shifted right c units.
Here is an example using a polynomial.
This is the graph of a function f ^ xh =- x 2 - 4x
y
5
4
3
2
1
-5
-4
-3
-2 -1 0
-1
1
2
3
4
5
x
-2
-3
-4
f ^ xh
a
-5
Draw the graph for f ^ x - 3h .
b
` shift f ^ xh 3 units to the right
Draw the graph for g^ xh =-^ x + 1h2 - 4^ x + 1h .
This is simply f ^ x + 1h .
` shift f ^ xh 1 unit to the left
y
y
5
5
f ^ x + 1h
4
3
-5
-4
-3
3
2
2
1
1
-2 -1 0
-1
-2
1
2
3
4
5
x
-5
right 3 units
-4
-5
-4
-3
left 1 unit
-3
f ^ xh
4
-2 -1 0
-1
1
2
3
4
5
x
-2
-3
-4
f ^ x - 3h
-5
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f ^ xh
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Functions
Using Our Knowledge
Inside and Outside the Brackets
When shifting graphs, you don’t even need to know what the original function is. Just remember that:
Outside the bracket
•
•
If the constant is outside the bracket like f ^ xh ! c then the graph shifts vertically.
If the constant is inside the bracket like f ^ x ! ch then the graph shifts horizontally.
Inside the bracket
Here is an example which has both.
The graph below represents f ^ xh . Use it to draw the graph of f ^ x - 2h + 3
y
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1 0
-1
1
2 3
4
5
6
x
-2
-3
-4
-5
-6
The new graph f^ x - 2h + 3 has -2 inside the bracket and +3 outside the bracket. This means that:
•
•
Step 1: Shift the graph 2 units to the right
f ^ xh is shifted 2 units to the right
f ^ xh is shifted 3 units upwards
y
y
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1-10
-2
-3
-4
-5
-6
20
6
5
4
3
2
1
2 units
1 2 3 4
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Final graph: f^ x - 2h + 3
Step 2: Shift this graph 3 units upwards
TOPIC
5 6
x
-6 -5 -4 -3 -2 -1-10
-2
-3
-4
-5
-6
y
6
5
4
3
2
1
3 units
1 2 3 4
5 6
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x
-6 -5 -4 -3 -2 -1-10
-2
-3
-4
-5
-6
1 2 3 4
5 6
x
Functions
Questions
Using Our Knowledge
1. The graph below is of f ^ xh = 2x + 1 . Draw these graphs on the same set of axes:
f ^ x + 2h
a
f^ xh
4
3
f ^ xh + 2
b
y
2
1
f ^ x - 3h
c
-4
-3
-2
-1
0
1
2
3
4
x
-1
-2
f ^ xh - 3
d
-3
-4
2. The graph below represents g^ xh . Use it to draw g^ x + 4h on the other set of axes.
y
-5
-4
-3
-2
-1
y
5
5
4
4
3
3
2
2
1
1
0
-1
1
2
3
4
5
6
x
-5
-4
-3
-2
-1
0
-1
-2
-2
-3
-3
-4
-4
-5
-5
1
2
3
4
5
6
x
3. Explain the difference between the following graphs, if f ^ xh is any function.
a
f ^ x + 2h and f ^ x - 2h
b
f ^ x + 4h and f ^ xh + 4
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Functions
Questions
Using Our Knowledge
4. The graph below is of f ^ xh =- x3 - 2x 2 + 5x + 6 . Draw these graphs on the same set of axes:
y
10
9
8
7
6
5
4
3
2
1
-11 -10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
1
2
-2
-3
-4
-5
-6
-7
-8
-9
-10
a
f ^ x + 4h
b
f ^ xh - 4
c
g^ xh =-^ x - 2h3 - 2^ x - 2h2 + 5^ x - 2h + 6
22
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4
5
6
7
8
9
10
11
x
Functions
Questions
Using Our Knowledge
5. The graph below is of f ^ xh . Use it to draw f ^ x + 1h - 3 on the other set of axes:
y
y
4
4
3
3
2
2
f^ xh
1
-4
-3
-2
-1
0
1
2
1
3
4
x
-4
-3
-2
-1
0
-1
-1
-2
-2
-3
-3
-4
-4
1
2
3
x
4
6. The graph on the right shows the function f ^ x + 5h + 7 . Draw the original f ^ xh on the left set of axes.
y
y
5
5
4
3
-5
-4
-3
-2
4
f^ x + 5h + 7
3
2
2
1
1
-1 0
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1 0
-1
-2
-2
-3
-3
-4
-4
-5
-5
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2
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x
23
Functions
Thinking More
Inverses, f -1 (x)
y = 2x and y = x have the inverse effect on any number. Each one is the inverse of the other.
2
An inverse function reverses the effect of the original function.
x
1st input
f ^ xh
1st output
2nd intput
f ^ xh
Inverse of f ^ xh
2nd output
x
st
Same as 1 input
For example:
5
1st input
2x
This means "inverse", not
1st output
2nd intput
10
1
f ^ xh
The inverse of f ^ xh is written as f -1 ^ xh .
The easiest way to find an inverse is to switch the pronumerals and then solve for y.
Find the inverse of these functions:
a
f ^ xh = 2x
` y = 2x
b
Replace f ^ xh with y
` y = 3x - 1
For inverse: x = 2y
Switch x and y
For inverse: x = 3y - 1
`y= x
2
Solve for y
` y = x+1
3
` g -1 ^ x h = x + 1
3
` f -1 ^ x h = x
2
c
f ^ x h = x3 - 1
d
` y = x3 - 1
For inverse: x = y3 - 1
` f -1 ^ x h = 3 x + 1
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p^ x h = 4
x
`y= 4
x
For inverse: x = 4
y
`y= 4
x
-1
` p ^ xh = 4
x
` y =3 x+1
24
g ^ x h = 3x - 1
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x
2
2nd output
5
Functions
Questions
Thinking More
1. Answer these questions about these functions:
(i)
f ^ xh = 5x - 10
(ii) g ^ xh = x + 1
4 2
(iii) h^ xh = 4x - 2
(iv) m^ xh = x + 2
5
a
Find f ^2h .
d
Find g^-1h .
b
Substitute this value into h^ xh and m^ xh .
e
Substitute this value into h^ xh and m^ xh .
c
Is h^ xh or m^ xh the inverse of f ^ xh .
f
Is h^ xh or m^ xh the inverse of g^ xh .
2. Match each function to its inverse.
1
f ^ x h = 3x + 2
a
f -1 ^ xh =- x + 2
6 3
2
f ^ xh = 10x - 5
b
f -1 ^ x h =
3
f ^ xh =-6x + 4
c
f -1 ^ x h = x - 2
3
4
f ^ x h = x2 + 3
d
f -1 ^ x h = x + 1
2 10
5
f ^ x h = x3 - 2
e
f -1 ^ x h = 3 x + 2
6
f ^ x h = x2 - 3
f
f -1 ^ x h = x - 2
6 3
7
f ^ xh = 2x - 1
5
g
f -1 ^ x h =
8
f ^ x h = x3 + 2
h
f -1 ^ x h = x + 1
10 2
9
f ^ xh = 6x + 4
i
f -1 ^ x h = 3 x - 2
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x-3
x+3
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Functions
Questions
3. Find the inverse of each of these functions:
a
a^ xh = 4x
b
b^ xh = 2 - 4x
c
c ^ x h = 3x + 2
7
d
d ^ xh = 4 x - 3
5
2
e
e^ xh = 3 x + 1
f
f ^ xh = 3
x
g
g^ x h =
h
h^ xh = mx + c
26
3
x+1
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Thinking More
Functions
Thinking More
Inverse Graphs and Inverse Functions
Here are the graphs of f ^ xh = 2x - 3 and f -1 ^ xh = x + 3 on the same set of axes:
2
y
5
4
3
2
f
-5
-4
-x
-3
y=x
^ xh
1
-2
-1 0
-1
1
2
3
4
x
5
f ^ xh
-2
-3
-4
-5
As you can see, the graph of f -1 ^ xh is simply the reflection of f ^ xh around the line y = x . This makes sense since
the inverse was found by switching x and y in the equation.
This is always the case.
To draw any inverse f -1 ^ xh , simply find the reflection of f ^ xh around the line y = x . Here is another example:
The graph below is of f ^ xh = x 2 - 4x + 3 . Find the graph of f -1 ^ xh :
y
y
5
5
4
4
3
3
2
2
1
-3
-2
0
-1
-1
1
2
3
4
5
x
Flip f ^ xh
around y = x
1
-3
-2
0
-1
1
2
3
4
5
x
-1
-2
-2
-3
-3
-4
-4
-5
-5
f -1 ^ xh will always intersect with f ^ xh over the line y = x .
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Thinking More
Using the vertical line test, it’s easy to see that the inverse of a parabola is not a function, but the inverse of a straight
line is a function. This means that an inverse isn’t always an inverse function. When do inverse functions exist?
Remember
a function is a relation which has only one output value for each input value.
Functions can be divided into two main types:
•
Many-to-one functions: Although each input value must only have
one output value, the same output value could come from more
than one (many) input values.
•
Input values
Output values
Input values
Output values
One-to-one functions: Each output value comes from a different
input value.
f -1 ^ xh will only be a function if it is one-to-one. If so, then f -1 ^ xh will also be one-to-one.
A simple test to determine whether or not a function 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 function.
•
If no horizontal line can cut the graph more than once then the graph is a one-to-one function.
Test whether f ^ xh and g ^ xh below will have an inverse functions.
y
-5
-4
-3
-2
y
5
5
4
4
3
3
2
2
1
1
-1 0
-1
1
2
3
4
5
x
-5
SERIES
TOPIC
-2
-1 0
-1
-2
-3
-3
-4
-4
-5
-5
` f -1 ^ xh will be a function.
K 14
-3
-2
This function is one-to-one since no horizontal
line cuts the graph twice
28
-4
2
3
4
5
x
This function is many-to-one since there is a
horizontal that cuts the graph more than once.
` g-1 ^ xh will not be a function.
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Thinking More
As usual, to find the inverses of f ^ xh and g^ xh on the previous page, just reflect the graph around the line y = x :
y
-4
-3
-2
y
5
5
4
4
3
3
2
2
1
1
-1 0
-1
1
2
3
4
5
x
-4
-3
-2
-1 0
-1
-2
-2
-3
-3
-4
-4
-5
-5
1
2
3
4
5
x
The vertical line test can be used on the graphs of f -1 ^ xh and g-1 ^ xh to test whether or not these are functions.
Here are some important points to remember about functions:
1. A relation is an expression involving two variables ( x and y ) and the equals (=) sign.
2. Functions are relations that have only one output for every input.
3. In functions, it is possible for many inputs to have the same output. It is impossible to have many outputs
for a single input.
4. Functions can be given names like f ^ xh or g^ xh . This is called function notation.
5. The vertical line test is used (on a graph) to test whether or not a relation is a function.
6. The domain of a function is the set of allowed x-values (input values).
7. The range of a function is the set of y-values (output values).
8. The graph of f ^ xh ! c is just the graph of f ^ xh shifted up (+) or down (-).
9. The graph of f ^ x ! ch is just the graph of f ^ xh shifted left (+) or right (-).
10. The inverse of a function f ^ xh is written as f -1 ^ xh
11. f -1 ^ xh will only be a function if f ^ xh is a one-to-one function.
1
12. ^ f -1h- ^ xh = f ^ xh . The inverse, of an inverse is the original function.
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4. Identify if these graphs represent a one-to-one or many-to-one function and state whether or not its
inverse is a function.
y
a
b
y
x
x
y
c
d
y
x
x
y
e
y
f
x
30
K 14
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x
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Thinking More
5. Sketch the inverse of these graphs on the same axes. State whether or not the inverse is a function.
y
a
-4
-3
-2
-1
4
4
3
3
2
2
1
1
0
1
2
3
4
x
-2
-1
-3
-2
-1
0
-1
-2
-2
-3
-3
-4
-4
y
-3
-4
-1
c
-4
y
b
4
3
3
2
2
1
1
1
2
3
4
x
-4
-3
-2
-1
0
-1
-1
-2
-2
-3
-3
-4
-4
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3
4
1
2
3
4
x
y
d
4
0
1
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x
31
Functions
Questions
y
e
-4
-3
-2
-1
Thinking More
y
f
4
4
3
3
2
2
1
1
0
1
2
3
4
x
-4
-3
-2
-1
-2
-2
-3
-3
-4
-4
y
4
3
2
1
-3
-2
-1
0
1
2
3
4
x
-1
-2
-3
-4
32
K 14
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0
-1
6. Use the graph of f ^ xh = 3 below to draw f -1 ^ xh . What do you notice?
x
-4
-1
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Functions
Answers
Basics:
Knowing More:
1. A function assigns a single output value to
a single input value, whereas a relationship
can have multiple out put values for a single
input value.
2. a
c
1#x16
c
11x#6
d
1#x#6
e
-5 1 x # 10
f
x 2 -3
g
x17
h
x#7
g
No
d
Yes
h
i
19
j
4. a 234
d
g
d
3 - 4n
3 - 8t
b
-33
c
22
80
e
-53
f
8m2 + 4m - 3
-4
h
x ! 6-2, 8h
b
Yes
c
d
3. a 1 1 x 1 6
b
b
x ! ^-2, 8 @
x ! 61, 5 @
x ! ^-3, 2 @
2. a Yes
3. a -13
b
f
e
c
x ! ^1, 5h
x ! ^2, 3h
x ! ^-3, -1 @ , ^3, 3h
x ! ^-3, -1h , 63, 3h
x ! ^-5, -1 @ , 63, 8 @
x ! 6-4, 3h , ^7, 3h
x =-3 or x =-1
5. a Function
b
Function
i
-3 1 x # 4 or 5 # x 1 8
c
Function
d
Not a function
j
2 # x 1 6 or 10 1 x # 20
e
Not a function
f
Function
k
0 1 x # 6 or
l
x 1 4 or
Knowing More:
1. a
c
x$8
4. a x can be any number except 0
x 1 3 means that x is less than 3.
3 is excluded.
x # 3 means that x is less than OR
equal to 3. 3 is included.
b
x17
3 means 'infinity' and - 3 means
'negative infinity'.
x ! ^-2, 2h means that x is between -2
and 2 but is neither -2 nor 2.
-2 and 2 are excluded.
x ! 6-2, 2 @ means that x is between -2
and 2 and can be -2 or 2.
-2 and 2 are included.
b
x can be any number except 2
c
x can be any number except 0
d
x can be any number
e
x can only be positive or 0
f
x can only be greater than or equal to 4.
x $ 4.
g
x can only be greater than or equal
to -3. x $ -3 .
h
x can only be greater than or equal to 1 .
2
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Functions
Answers
Knowing More:
4.
Knowing More:
i
x can be any number except -1 or 1
j
x can be any number except 0 or 3.
k
x can be any number except -2 or 2
l
x can only be negative or 0.
7. e
x ! ^-3, 3h
f
x ! 6-2, 3h
h
y ! 62, 3h
y ! 60, 3h
g
x ! 60, 3h
y ! 60, 3h
x ! ^-3, 3h
y ! ^0, 3h
5. a -3 1 x 1 -3 or -3 1 x 1 3
8. a
b
c
x2 + y2 = 9
x$1
b
Maximum value of y is 3
Minimum value of y is -3
c
-3 # y # 3
d
Maximum value of x is 3
Minimum value of x is -3
e
-3 # x # 3
-3 1 x 1 -2 or
-2 1 x 1 7 or
71x13
d
-3 1 x 1 -4 or
-4 1 x 1 5 or
51x13
e
x#1
f
x 2 -3
6. a
b
c
d
e
9. a Highest point is at ^1, 4h
Lowest points are at ^-2, -5h and ^4, -5h
x ! 60, 3h
x ! ^-3, 0 @
c
-5 # y # 4
1.
x ! ^-3, -8h , ^-8, 5h , ^5, 3h
a
x ! ^-3, -4h , ^-4, -3h , ^-3, 3h
x ! 8- 2 , 3j
3
7. a
x ! ^-3, 3h
b
x ! ^-3, 3h
d
y ! ^-3, -2 @
K 14
SERIES
TOPIC
x ! ^-3, 3h
y ! 6-1, 3h
y ! ^-3, 3h
34
-2 # x # 4
Using Our Knowledge:
x ! ^0, 3h
f
c
b
x ! ^-3, 3h
y ! 6-1, 1 @
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b
d
c
Functions
Answers
Using Our Knowledge:
Using Our Knowledge:
2.
5.
f^ x + 1h - 3
3. a
b
f ^ x + 2h is f ^ xh shifted two units to
the left, whereas f ^ x - 2h is f ^ xh
shifted two units to the right.
6.
f ^ x + 4h is f ^ xh shifted 4 units to the
left, whereas f ^ xh + 4 is f ^ xh shifted
4 units upwards.
4.
a
c
b
f^ xh
Thinking More:
1. a
b
a
b
c
f ^2h = 0
h^0h =-2
m^0h = 2
f ^ xh is shifted 4 units to the left
c
f ^ xh is shifted 4 units downwards
m^ xh is the inverse of f ^ xh
The graph of g^ xh is the graph of
f ^ xh shifted 2 units to the right
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Thinking More:
Thinking More:
1. d
g^-1h = 1
4
e
h` 1 j =-1
4
f
h^ xh is the inverse of g^ xh
2. 1
2
3
4
5
6
7
8
9
5. a
m ` 1 j = 41
4
20
f ^ x h = 3x + 2
f ^ xh = 10x - 5
f ^ xh =-6x + 4
f ^ x h = x2 + 3
f ^ x h = x3 - 2
f ^ x h = x2 - 3
f ^ xh = 2x - 1
5
f ^ x h = x3 + 2
f ^ xh = 6x + 4
c
h
a
b
e
f -1 ^ x h = x - 2
3
x
-1
f ^ xh =
+1
10 2
f -1 ^ xh =- x + 2
6 3
f -1 ^ x h =
x-3
f -1 ^ x h = 3 x + 2
-1
^ xh =
g
f
d
f -1 ^ x h = x + 1
2 10
i
f
f -1 ^ x h = 3 x - 2
f -1 ^ x h = x - 2
6 3
b
b-1 ^ xh = x - 2
-4
c
c-1 ^ xh = 7x - 2
3
d
d-1 ^ xh = 5x + 15
4
8
e
e-1 ^ xh = ^ x - 1h3
f
g-1 ^ xh = 3 - 1
x
h
g
b
x+3
a-1 ^ xh = x
4
3. a
The inverse is a function
f
-1
The inverse is a function
^ xh = 3
x
c
h-1 ^ xh = x - c
m
4. a Its inverse will be a function.
36
b
Its inverse will be a function.
c
Its inverse will be a function.
d
Its inverse will not be a function.
e
Its inverse will not be a function.
f
Its inverse will not be a function.
K 14
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The inverse is not a function
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Thinking More:
Thinking More:
5. d
6.
The inverse is a function
The graph of the inverse is the same as the
graph of the function.
e
The inverse is not a function
f
The inverse is a function
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