1) The document discusses various topics in mathematics including sets, functions, composite functions, exponential and logarithmic graphs, and graph transformations.
2) It provides definitions and examples of sets, functions, and how to represent functions using formulas, arrow diagrams, and graphs. Composite functions are defined as functions of other functions.
3) The document explains how to graph exponential and logarithmic functions and describes the key features of these graphs. It also discusses how different transformations can move a graph in various ways, such as reflecting it or stretching/squashing it.
This learner's module will discuss or talk about the Graph of Quadratic Functions. It will also discuss on how to draw the Graph of Quadratic Functions using the vertex, axis of symmetry, etc.
This learner's module will discuss or talk about the Graph of Quadratic Functions. It will also discuss on how to draw the Graph of Quadratic Functions using the vertex, axis of symmetry, etc.
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
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See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
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👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
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Here’s what you’ll gain:
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We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
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Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
DevOps and Testing slides at DASA ConnectKari Kakkonen
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Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
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After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
1. www.mathsrevision.com
Higher
Higher Unit 1
Outcome 2
What is a set
Recognising a Function in various formats
Composite Functions
Exponential and Log Graphs
Graph Transformations
Trig Graphs
Connection between Radians and degrees
Trig Exact Values
Basic Trig Identities
Exam Type Questions
www.mathsrevision.com
2. Sets & Functions
Higher
Outcome 2
www.mathsrevision.com
Notation & Terminology
SETS: A set is a collection of items which have
some common property.
These items are called the members or
elements of the set.
Sets can be described or listed using
“curly bracket” notation.
3. Sets & Functions
Outcome 2
Higher
www.mathsrevision.com
eg
{colours in traffic lights}
DESCRIPTION
eg
{square nos. less than 30}
= {red, amber, green}
LIST
= { 0, 1, 4, 9, 16,
25}
NB: Each of the above sets is finite because we
can list every member
4. = {1, 2, 3, 4, ……….}
Sets & Functions
Outcome 2
Higher
www.mathsrevision.com
We can describe numbers by the following sets:
N = {natural numbers}
W = {whole numbers} = {0, 1, 2, 3, ………..}
Z = {integers}
= {….-2, -1, 0, 1, 2, …..}
Q = {rational numbers}
This is the set of all numbers which can be written
as fractions or ratios.
eg
5 = 5/1
-7 =
-7
55
/1
0.6 = 6/10 = 3/5
11
5. www.mathsrevision.com
Higher
Sets & Functions
Outcome 2
R = {real numbers}
This is all possible numbers. If we plotted values
on a number line then each of the previous sets
would leave gaps but the set of real numbers would
give us a solid line.
We should also note that
N
“fits inside”
W
W “fits inside”
Z
Z
Q
Q
“fits inside”
“fits inside”
R
6. Sets & Functions
Outcome 2
Higher
www.mathsrevision.com
N
W
Z
Q
R
When one set can fit inside another we say
that it is a subset of the other.
The members of R which are not inside Q are called
irrational numbers. These cannot be expressed as
fractions and include π , √2, 3√5 etc
7. Sets & Functions
Outcome 2
www.mathsrevision.com
Higher
To show that a particular element/number
belongs to a particular set we use the symbol ∈.
eg
3 ∈ W but
0.9 ∉ Z
Examples
{ x ∈ W: x < 5 } =
{ x ∈ Z: x ≥ -6 }
{ x ∈ R: x2 = -4 }
=
{ 0, 1, 2, 3, 4 }
{ -6, -5, -4, -3, -2,
…….. }
=
{ } or Φ
This set has no elements and is called the empty set.
8. Functions & Mappings
www.mathsrevision.com
Higher
Outcome 2
Defn: A function or mapping is a relationship between
two sets in which each member of the first set
is connected to exactly one member in the second set.
If the first set is A and the second B then we often write
f: A → B
The members of set A are usually referred to as the
domain of the function (basically the starting values or
even x-values) while the corresponding values or images
come from set B and are called the range of the function
(these are like y-values).
9. Functions & Mapping
Outcome 2
Higher
www.mathsrevision.com
Functions can be illustrated in three ways:
1) by a formula.
2) by arrow diagram.
3) by a graph
Example
FORMULA
(ie co-ordinate diagram).
Suppose that
f(x) = x2 + 3x
then f(-3) = 0 ,
f(1) = 4
f: A → B
is defined by
where A = { -3, -2, -1, 0, 1}.
f(-2) = -2 , f(-1) = -2 ,
f(0) = 0 ,
NB: B = {-2, 0, 4} = the range!
11. Functions & Graphs
Outcome 2
Higher
www.mathsrevision.com
In a GRAPH we get :
NB:
This graph consists of 5 separate points.
It is not a solid curve.
12. Functions & Graphs
Outcome 2
Higher
www.mathsrevision.com
A
B
a
b
c
d
Recognising Functions
e
f
g
Not a function
two arrows leaving b!
A
a
b
c
d
B
e
f
g
YES
13. Functions & Graphs
Outcome 2
Higher
www.mathsrevision.com
A
B
a
b
c
d
e
f
g
Not a function - d unused!
A
a
B
e
b
c
f
d
g
YES
14. Higher
Functions & Graphs
Outcome 2
www.mathsrevision.com
Recognising Functions from Graphs
If we have a function f: R → R (R - real nos.)
then every vertical line we could draw would cut
the graph exactly once!
This basically means that every x-value has one,
and only one, corresponding y-value!
19. Composite Functions
Outcome 1
www.mathsrevision.com
Higher
COMPOSITION OF FUNCTIONS
( or functions of functions )
Suppose that f and g are functions where
f:A → B
with
and
f(x) = y
where
x∈ A,
g:B → C
and
y∈ B
g(y) = z
and z∈ C.
Suppose that h is a third function where
h:A → C
with
h(x) = z .
25. Composite Functions
Outcome 1
x=0
(ii) Suppose that g(x) = √(x2 + 2x - 8)
(0 + 4)(0 - 2)
x=3
= 8) ≥ 0
We need (x2 + 2x -negative
(3 + 4)(3 - 2)
x = -5
= positive
Suppose (x2 + 2x - 8) = 0
(-5 + 4)(-5 - 2)
= positive Then (x + 4)(x - 2) = 0
www.mathsrevision.com
Higher
So x = -4 or x = 2
-4
2
Check values below -4 , between -4 and 2, then above 2
So
domain = { x∈R: x ≤ -4 or x ≥ 2 }
26. Exponential (to the power of)
Graphs
Outcome 1
www.mathsrevision.com
Higher
Exponential Functions
A function in the form
f(x) = ax
where a > 0, a ≠ 1
is called an exponential function to base a .
Consider f(x) = 2x
x
f(x)
-3
-2
-1
0
1
2
3
/8
¼
½
1
2
4
8
1
27. Graph
www.mathsrevision.com
Higher
Outcome 1
The graph is like
y = 2x
(0,1)
(1,2)
Major Points
(i) y = 2x passes through the points (0,1) & (1,2)
(ii) As x ∞ y ∞ however as x ∞ y 0 .
(iii) The graph shows a GROWTH function.
29. Graph
www.mathsrevision.com
Higher
The graph is like
(2,1)
Outcome 1
(1,0)
y = log2x
NB: x > 0
Major Points
(i) y = log2x passes through the points (1,0) & (2,1) .
(ii) As x y but at a very slow rate and as x 0 y
- .
30. Exponential (to the power of)
Graphs
Outcome 1
www.mathsrevision.com
Higher
The graph of y = ax always passes through (0,1) & (1,a)
It looks like ..
Y
y = ax
(1,a)
(0,1)
x
33. Graph of -f(x) Transformations
Outcome 1
www.mathsrevision.com
Higher
y = f(x)
y = x2
y = -f(x)
y = -x2
Mathematically
y = –f(x)
reflected f(x) in the x - axis
34. Graph of -f(x) Transformations
Outcome 1
www.mathsrevision.com
Higher
y = f(x)
y = 2x + 3
y = -f(x)
y = -(2x + 3)
Mathematically
y = –f(x)
reflected f(x) in the x - axis
35. Graph of -f(x) Transformations
Outcome 1
www.mathsrevision.com
Higher
y = f(x)
y = x3
y = -f(x)
y = -x3
Mathematically
y = –f(x)
reflected f(x) in the x - axis
36. Graph of f(-x) Transformations
Outcome 1
www.mathsrevision.com
Higher
y = f(x)
y = f(-x)
y = x + 2
y = -x + 2
Mathematically
y = f(-x)
reflected f(x) in the y - axis
37. Graph of f(-x) Transformations
Outcome 1
www.mathsrevision.com
Higher
y = f(x)
y = (x+2)2
y = f(-x)
Mathematically
y = f(-x)
reflected f(x) in the y - axis
y = (-x+2)2
38. Graph of f(x) ± k Transformations
Outcome 1
www.mathsrevision.com
Higher
y = f(x)
y = f(x) ± k
y = x2
y = x2-3
y = x2 + 1
Mathematically
y = f(x) ± k
moves f(x) up or down
Depending on the value of k
+ k move up
- k move down
39. Graph of f(x ± k) Transformations
Mathematically
y = f(x ± k)
Higher
Outcome 1
www.mathsrevision.com
moves f(x) to the left or right
2
y = f(x)
y = x
depending on the value of k
-k move right
+ k move left
y = f(x ± k)
y = (x-1)2
y = (x+2)2
40. Graph of k f(x) Transformations
Outcome 1
www.mathsrevision.com
Higher
y = f(x)
y = k f(x)
y = x2-1
y = 2(x2-1)
y = 0. 5(x2-1)
Mathematically
y = k f(x)
Multiply y coordinate by a factor of k
k > 1 (stretch in y-axis direction)
0 < k < 1 (squash in y-axis direction)
41. Graph of -k f(x) Transformations
Outcome 1
www.mathsrevision.com
Higher
y = f(x)
y = x2-1
y = k f(x)
y = -2(x2-1)
y = -0. 5(x2-1)
Mathematically
y = -k f(x)
k = -1 reflect graph in x-axis
k < -1 reflect f(x) in x-axis & multiply by a
factor k (stretch in y-axis direction)
-1 < k < 0 reflect f(x) in x-axis multiply by a
factor k (squash in y-axis direction)
42. Graph of f(kx) Transformations
Outcome 1
www.mathsrevision.com
Higher
y = f(x)
y = f(kx)
y = (x-2)2
y = (2x-2)2
y = (0.5x-2)2
Mathematically
y = f(kx)
Multiply x – coordinates by 1/k
k > 1 squashes by a factor of 1/k in the x-axis direction
k < 1 stretches by a factor of 1/k in the x-axis direction
43. Graph of f(-kx) Transformations
Outcome 1
www.mathsrevision.com
Higher
y = f(x)
y = (x-2)2
y = f(-kx) y = (-2x-2)2
y = (-0.5x-2)2
Mathematically
y = f(-kx)
k = -1 reflect in y-axis
k < -1 reflect & squashes by factor of 1/k in x direction
-1 < k < 0 reflect & stretches factor of 1/k in x direction
44. Trig Graphs
Higher
Outcome 1
www.mathsrevision.com
The same transformation rules
apply to the basic trig graphs.
NB: If f(x) =sinx°
then 3f(x) = 3sinx°
and
f(5x) = sin5x°
Think about sin replacing f !
Also if g(x) = cosx° then g(x) –4 = cosx ° –4
and g(x+90) = cos(x+90) °
Think about cos replacing g !
52. Radians
Higher
Outcome 1
www.mathsrevision.com
Since we have a semi-circle the angle must be 180 o.
We now get a simple connection between degrees and radians.
π (radians) = 180o
This now gives us
2π = 360o
π /2 = 90o
3π /2 = 270o
π /3 = 60o
2π /3 = 120o
π /4 = 45o
3π /4 = 135o
π /6 = 30o
5π /6 = 150o
NB: Radians are usually expressed as fractional multiples of π.
55. Converting
Outcome 1
Higher
www.mathsrevision.com
Example 5
Angular Velocity
In the days before CDs the most popular format for
music was “vinyls”.
Singles played at 45rpm while albums played at 331/3 rpm.
rpm =revolutions per minute !
Going back about 70 years an earlier version of vinyls
played at 78rpm.
Convert these record speeds into “radians per second
56. Converting
Outcome 1
www.mathsrevision.com
Higher
NB:
So
So
1 revolution = 360o = 2π radians 1 min = 60 secs
45rpm = 45 X 2π or 90π radians per min
90π
=
/60 or 3π/2 radians per sec
331/3rpm = 331/3 X 2π or 662/3 π radians per min
=
So
662/3 π /60
or
10 π
/9 radians per sec
78rpm = 78 X 2π or 156π radians per min
=
/60
15π
or
13 π
/5 radians per sec
57. Exact value table
and quadrant rules.
Outcome 1
www.mathsrevision.com
Higher
tan150o
= tan(180-30) o
= -tan30o
=
-1
/√3
(Q2 so neg)
cos300o
= cos(360-60) o = cos60o
= 1/2
= sin(180-60) o = sin60o
= √ 3/2
= tan(360-60)o = -tan60o
= -√3
(Q4 so pos)
sin120o
(Q2 so pos)
tan300o
(Q4 so neg)
58. Exact value table
and quadrant rules.
Outcome 1
www.mathsrevision.com
Higher
Find the exact value of
cos2(5π/6) – sin2(π/6)
cos(5π/6) = cos150o = cos(180-30)o = -cos30o
= - √3 /2
(Q2 so neg)
sin(π/6)
= sin30o = 1/2
cos2(5π/6) – sin2(π/6)
= (-
√3
/2)2 – (1/2)2 = ¾ - 1/4 = 1/2
60. Trig Identities
Outcome 1
www.mathsrevision.com
Higher
An identity is a statement which is true for all values.
eg
3x(x + 4) = 3x2 + 12x
eg
(a + b)(a – b) = a2 – b2
Trig Identities
(1)
sin2θ + cos2 θ = 1
(2)
sin θ = tan θ
cos θ
θ ≠ an odd multiple of π/2 or 90°.
63. Trig Identities
Outcome 1
Higher
sin θ = 5/13
www.mathsrevision.com
Example1
where 0 < θ < π/2
Find the exact values of cos θ and tan θ .
cos2 θ = 1 - sin2 θ
Since θ is between 0 < θ < π/2
= 1 – (5/13)2
then cos θ > 0
= 1–
So
=
144
25
/169
/169
cos θ = √(144/169)
=
12
/13 or -12/13
tan θ = sinθ =
cos θ
cos θ = 12/13
5
/13 ÷ 12/13
=
5
/13
tan θ =
5
/12
X
13
/12
64. Trig Identities
Outcome 1
www.mathsrevision.com
Higher
Given that cos θ = -2/ √ 5
Find sin θ and tan θ.
2
= 1 – ( /√5 )
-2
2
= 1 – 4/ 5
=
1
Hence sinθ =
=
sin θ = √( /5)
1
-1
/2
/√5
3 π
/2
sinθ < 0
tan θ = sinθ =
cos θ
/5
= 1/ √ 5 or
3 π
Since θ is between π< θ <
sin θ = 1 - cos θ
2
where π< θ <
- 1
/√5
/ √ 5 ÷ -2/ √ 5
-1
-1
/√5
X
- √5
Hence tan θ = 1/2
/2
66. Graphs & Functions
Higher
The following questions are on
Graphs & Functons
Non-calculator questions will be indicated
You will need a pencil, paper, ruler and rubber.
Click to continue
67. Graphs & Functions
Higher
The diagram shows the graph of a function f.
f has a minimum turning point at (0, -3) and a
point of inflexion at (-4, 2).
y = 2f(-x)
a) sketch the graph of y = f(-x).
b) On the same diagram, sketch the graph of y = 2f(-x)
a)
y
y = f(-x)
Reflect across the y axis
4
2
b)
Now scale by 2 in the y direction
-1
3
4
x
-3
-6
Previous
Quit
Quit
Hint
Next
68. Graphs & Functions
Higher
The diagram shows a sketch of part of
the graph of a trigonometric function
whose equation is of the form y = a sin(bx ) + c
1
2a
Determine the values of a, b and c
a is the amplitude:
a=4
b is the number of waves in 2π
1 in π
2 in 2 π
b=2
c is where the wave is centred vertically
c=1
Hint
Previous
Quit
Quit
Next
69. Graphs & Functions
Functions f ( x) =
Higher
1
and g ( x) = 2 x + 3 are defined on suitable domains.
x−4
a)
Find an expression for h(x) where h(x) = f(g(x)).
b)
Write down any restrictions on the domain of h.
a)
b)
2x −1 ≠ 0
1
→
2x + 3 − 4
f ( g ( x)) = f (2 x + 3)
→ x≠
1
→ h( x ) =
2x −1
1
2
Hint
Previous
Quit
Quit
Next
70. Graphs & Functions
a) Express
Higher
f ( x) = x 2 − 4 x + 5 in the form ( x − a ) 2 + b
(2, 1)
b) On the same diagram sketch
y=f(x)
the graph of y = f ( x)
i)
ii)
b)
5
the graph of y = 10 − f ( x )
y= 10 - f x)
(
c) Find the range of values of x for
(2, 1)
which 10 − f ( x ) is positive
a)
c)
(2, -1)
( x − 2) 2 − 4 + 5 → ( x − 2) 2 − 4 + 5
Solve:
→ ( x − 2) 2 + 1
10 − ( x − 2) 2 − 1 = 0
→ ( x − 2) 2 = 9
→ ( x − 2) = ±3
y= -f(x)
→ x = −1 or 5
Hint
10 - f(x) is positive for -1 < x < 5
Previous
Quit
Quit
Next
71. Graphs & Functions
Higher
The graph of a function f intersects the x-axis at (–a, 0)
and (e, 0) as shown.
There is a point of inflexion at (0, b) and a maximum turning
point at (c, d).
Sketch the graph of the derived function
f′
m is +
m is +
m is -
f′(x)
Previous
Quit
Quit
Hint
Next
72. Graphs & Functions
Higher
Functions f and g are defined on suitable domains by f ( x) = sin( x) and g ( x ) = 2 x
a)
Find expressions for:
i)
f ( g ( x))
ii) g ( f ( x))
b)
Solve 2 f ( g ( x)) = g ( f ( x)) for 0 ≤ x ≤ 360
a)
f ( g ( x )) = f (2 x ) = sin 2x
b)
2sin 2 x = 2sin x
→ sin 2 x − sin x = 0
→ 2sin x cos x − sin x = 0
→ sin x = 0
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or
g ( f ( x)) = g (sin x) = 2sin x
cos x =
1
2
Quit
→ sin x(2 cos x − 1) = 0
x = 0°, 180°, 360°
x = 60°, 300°
Quit
Hint
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73. Graphs & Functions
Higher
The diagram shows the graphs of two quadratic
functions y = f ( x) and y = g ( x )
Both graphs have a minimum turning point at (3, 2).
Sketch the graph of
y = f ′( x)
y=f′(x)
and on the same diagram
sketch the graph of
y = g ′( x)
y=g′(x)
Hint
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74. Graphs & Functions
Functions
Higher
f ( x) = sin x, g ( x) = cos x and h( x) = x +
π
4
are defined on a suitable set of real numbers.
b) i)
ii)
g (h( x))
f (h( x ))
a) Find expressions for
f (h( x)) =
Show that
1
2
sin x +
1
2
cos x
Find a similar expression for g ( h( x))
and hence solve the equation
π
4
π
4
π
4
= sin( x + )
a)
f (h( x)) = f ( x + )
b)
sin( x + ) = sin x cos
π
4
f (h( x)) − g (h( x )) = 1 for 0 ≤ x ≤ 2π
g (h( x)) = cos( x + )
π
π
+ sin cos x
4
4
Now use exact values
Repeat for ii)
equation reduces to
Previous
2
sin x = 1
2
Quit
sin x =
Quit
2
1
=
2
2
x=
π 3π
,
4 4
Hint
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75. Graphs & Functions
Higher
A sketch of the graph of y = f(x) where ) = x 3 − 6 x 2 + 9 x
f (x
is shown.
The graph has a maximum at A and a minimum at B(3, 0)
a) Find the co-ordinates of the turning point at A.
g ( x = f ( the graph
b) Hence, )sketchx + 2) + 4 of
Indicate the co-ordinates of the turning points. There is no need to
calculate the co-ordinates of the points of intersection with the axes.
c) Write down the range of values of k for which g(x) = k has 3 real roots.
b)
c)
Differentiate
f ′( x) = 3 x 2 − 12 x + 9
when x = 1
a)
y=4
Graph is
t.p. at A is:
for SP, f′(x) = 0
x = 1 or x = 3
(1, 4)
moved 2 units to the left, and 4 units up
t.p.’s are:
(3, 0) → (1, 4)
(1, 4) → ( −1, 8)
For 3 real roots, line y = k has to cut graph at 3 points
Hint
from the graph, k ≥ 4
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76. Graphs & Functions
f ( x) = 3 − x
and
Higher
3
x
g ( x) = ,
x≠0
a) Find p( x) where p( x) = f ( g ( x ))
3
q( x) =
, x ≠ 3 find p(q ( x )) in its simplest form.
b) If
3− x
a)
p ( x) = f ( g ( x)) = f
p (q ( x)) =
b)
→
p 3 =
÷
3− x
3
÷
x
3
3
−1÷
3− x
3
3− x
9 − 3(3 − x) 3 − x
÷×
3− x 3
Previous
→
→
Quit
3−
3
x
→
3 x −3
x
3( x −1)
x
→
9
3
→
− 3 ÷÷
3− x
3− x
3x 3 − x
×
3− x
3
Quit
→
x
Hint
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77. Graphs & Functions
Higher
Part of the graph of y = f ( x ) is shown in the diagram.
On separate diagrams sketch the graph of
y = f ( x + 1) a)
y = −2 f ( x) b)
Indicate on each graph the images of O, A, B, C, and D.
a)
b)
graph moves to the left 1 unit
graph is reflected in the x axis
graph is then scaled 2 units in the y direction
Hint
Previous
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78. Graphs & Functions
Higher
Functions f and g are defined on the set of real numbers by
f ( x) = x − 1 and g ( x) = x 2
a) Find formulae for
i)
f ( g ( x))
ii) g ( f ( x))
b) The function h is defined by h( x) = f ( g ( x)) + g ( f ( x))
2
Show that h( x) = 2 x − 2 x and sketch the graph of h.
c) Find the area enclosed between this graph and the x-axis.
a)
b)
h( x) = x − 1 + ( x − 1)
g ( f ( x)) = g ( x − 1) = ( x − 1)
f ( g ( x )) = f ( x 2 ) = x 2 − 1
c)
2
2
h( x ) = x 2 − 1 + x 2 − 2 x + 1
Graph cuts x axis at 0 and 1
Area
Previous
=
1
3
unit
Now evaluate
∫
1
0
2
→ 2x2 − 2x
2 x 2 − 2 x dx
Hint
2
Quit
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79. Graphs & Functions
Higher
The functions f and g are defined on a suitable domain by
f ( x) = x 2 − 1 and g ( x) = x 2 + 2
b) Factorise f ( g ( x ))
a) Find an expression for f ( g ( x ))
a)
f ( g ( x)) = f ( x + 2) = ( x + 2 ) − 1
b)
Difference of 2 squares
2
Simplify
2
2
→
(( x
2
)(( x
+ 2) + 1
2
)
+ 2) −1
→ ( x 2 + 3) ( x 2 + 1)
Hint
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